On the structure of theGL 2 of a ring

P. Cohn

doi:10.1007/BF02684355

On the structure of theGL 2 of a ring

Cohn, P. 2007-08-04 00:00:00 by P. M. COHN CONTENTS ]3Attt~S I, INTRODUCTION ............................................................................ 5 2. Elementary matrices over an arbitrary ring ................................................. 7 3, Direct and free. products of GlE-rings ....................................................... 1o 4" GL~(R) for a local ring ................................................................... [4 5- Discretely normed rings ................................................................... t 5 6. Rings of algebraic integers and algebraic functions ........................................... 2 i 7- k-rlngs with a degree-function .............................................................. ~4 8. Discretely ordered rings ................................................................... ",6 9. The commutator quotient structure of GE:~(R) and E~(R) .................................... 3I Io. Generating sets and free products in GF~(R) ................................................ 36 x z. The construction of homomorphisms between general linear groups ............................ 38 1,,. The analysis of isomorphlsms of general linear groups ........................................ 4t REr~:a~z.'ccss .................................................................................. 53 x. Introduction. The general linear group over a field, its subgroups and automorphisms, have been studied fairly extensively, even when the field is skew (cf. [8] and the ret~rences given there), but little is known about the general linear groups over an arbiuary ring. Now in the case of fields, the starting point is the observation that every" invertible matrix is a product of elementary matrices; this suggests that in studying GL,(R) it is best to confine attention at first to rings which share this property, and we therefore define a generalized Euclidean ring, or GE-ring for short, as an integral domain (not necessarily commutative) such that for all n, GE, : Every invertible n × :~ matrix is a product q] elementary n × n matrices. Examples of GE-rings are (i) the classical Euclidean rings (cf. [17]), (ii) rings with a weak algorithm [5], in particular ti'ee associative algebra~ over a commutative field, and (iii) free products of GE-rings which are also semifirs (cf [6] and § 3 below); this includes in particular the group algebras of free groups and free products of skew fields. 365 6 P. M. COHN The present paper is a study of GL~(R), for various types of rings, and in particular GE-rings; as in the case of fields, there is a basic difference between the case of dimension 2 and dimension gleater than 2, so that it is reasonable to begin by concentrating attention on the former. For most of our results we shall find it enough to assume GE~ for n = 2 only; in fact we can even dispense with GE 2 by limiting ourselves to the subgroup of GL~(R) generated by the elementary matrices, and the results will usually be stated in this form. This more general point of view is actually forced on us when we come to give examples of rings that are not GE-rings. Compared with Euclidean rings, GE-rings have a more intrinsic definition, in that no norm function is involved. But for a closer study of GL~(R) we shall find some sort of norm function on R of great use. Most of our results will apply to discretely normed rings, defined in w 5 and to discretely ordered rings defined in w 8. It is the presence of the norm (or the ordering) which usually enables us to decide whether a given ring satisfies GE,. As an application we shall show that the ring of polynomials in any number of indeterminates with integer coefficients, and the ring of polynomials in at least two indeterminates with coefficients in a field, are not GE-rings (w 5). Further it is shown that the ring of algebraic integers in an imaginary quadratic number field is a GE-ring if and only if the field is Euclidean with respect to the usual norm (w 6). In particular (1) taking the integers in O (~--~) we obtain a principal ideal domain which is not a GE-ring (clearly this says rather more than the usual assertion that this ring is not Euclidean). The main tool of the paper is an explicit presentation of GL~(R) for suitable GE-rings, called universal GE-rings (w 3). As examples of such rings we have fields or more generally local rings (w 4); moreover, any GE-ring which is either discretely normed or discretely ordered is a universal GE-ring. This presentation brings out in a particularly clear form the extent to which GL~(R) is independent of the multiplicative structure of R, and which accounts for the crucial difference between GL~(R) for n = 2 and n>2. Thus a generalization of a ring homomorphism, the U-homomorphism, is introduced in w i I and it is shown that any U-homomorphism between two universal GE-rings induces a homomorphism between their GL~-groups. As a rather striking illustration one has the result that for any free associative algebra A over a commutative field F on at most countably many free generators, GL~(A)-~GL2(F[x]). The auto- morphisms of GL2(F[x]) constructed by Reiner in [16] can also be obtained very simply from this point of view. In the other direction the isomorphisms between the GL~-groups of GE-rings are studied under the assumption that the rings have a degree function defined on them such that all the elements of degree zero are units. It is shown that any such isomorphism can be built up by taking a U-isomorphism (or a U-anti-isomorphism), following it by a cen- (1) This answers the question raised by various authors (I. Reiner, J.-P. Serre) whether every Dedekind ring, or more particularly, every principal ideal domain, is a GE-ring. 366 ON THE STRUCTURE OF THE GL~ OF A RING tral homothety and an inner automorphism (just as in the case of fields, cf. [8], where the notion of U-(anti-)isomorphism reduces to the ordinary (anti-)isomorphism). The proof is based on the fact that in a ring of the type named, all dihedral subgroups of order eight are conjugate, in case the characteristic of the underlying field is ~ 2, while in charac- teristic 2 all subgroups of the type of the symmetric group of degree 3 are conjugate. This last fact rests on a surprisingly delicate argument (Lemma 12.3) and it would be of interest to have a simpler proof. The presentation introduced in w 2 can also be used to study the commutator quotient structure of GL2(R ). This is done in w 9 where an analogue of SLy(R) is defined, denoted by E2(R ) and it is shown that for a universal GE-ring, GL,(R)/E2(R)~--~U(R) a, where U(R) is the group of units of R and for any group G, G ~ = G/G' is G made abelian. Further, if R is a discretely normed ring in which U(R) is commutative, then l~.2(R)"-~ R/M, where M is a certain additive subgroup of R deter- mined by the units of R (for a precise statement see Theorem 9.3). A number of other applications and generalizations of known results are given in w Io. 2. Elementary matrices over an arbitrary ring. For a Euclidean ring R it is well known that GL2(R), the group of invertible 2 � 2 matrices over R, is generated by the elementary matrices (; ;) (; :) (o ;) where ~, ~, aeR and ~, [~ are units. Our object is to study, for any ring R, the group generated by these matrices; we shall denote this group by GE2(R ). In particular, we shall be interested in rings R for which GE2(R ) =GL2(R); such rings will be called GE2-rings. Clearly they include the GE-rings mentioned in w I. Let R be any ring (always associative, with I) ; generally we shall denote arbitrary elements of R by latin letters and reserve greek letters for invertible elements of R. The group of units of R is denoted by U(R) and Uo(R)-~U(R)u{o }. Further we set and denote by D=D2(R ) the group of 2� invertible diagonal matrices over R and by E=E2(R) the group generated by all E(a), aeR. Thus GE~(R) is the group generated by D and E. If we write Bq (a)= I-t-aeq, where e# are the usual matrix units, then Bx2(a)=E(--a)E(o) -1, S,l(a) =- E(o)-lE(a), and E(o) ~--~ B12(I) B21 (- I)BI~(I), E(a) = B12(-- a)E(o) ; these equations show that E2(R) is just the group generated by all B~(a) (aeR, i4:j). However, we shall mainly keep to the generators E(a), since the defining relations are 367 8 P. M. COHN expressed rather more easily in terms c t them. We have the following relations between the matrices (2. I) : (2.2) E(~)E(o)E(y) =--E(x +y), (2.3) E(=)E(~-I)E(~) = --D(e) x, yeR, (2.4) E(x) [cr [B] = [[3, ~]E(~-'x=), 0r ~eU(R). Their verification is elementary and is left to the reader. These three relations are basic for much of what follows, for as we shall see, there are large classes of rings in which the relations (2.2)-(2.4), together with the relations in D~(R), form a complete set ot defining relations for GE~(R). A ring which has this property is said to be universai for GE2; of particular interest are the GE2-rings which are universal for GE2, or the universal GE2-rings, as we shall call them. Thus a universal GE~-ring is characterized by the property that GL2(R ) is generated by the matrices (2. I), with (2.2-4) and the relations of D 2 as a complete set of defining relations. The first task is to derive a number of consequences of (2.2-4) : Putting x =y = 0 in (2.2) and ~=4-I in (2.3), we find (2.5) E(o)Z=--I, E(I) 3=-I, E(--I) 3=I- Putting y=--x in (2.2) and using (2.5) we get E(x) -1 ~- E(o)E(--x)E(o) ; (2.6) explicitly we have ~ -3' By (2.6) we have (2.7) E(x) E(y)-l= E(x--y) E(o)-l=--E(x--y)E(o). From (2.7) we easily obtain the following generalization of (2.2): (2. s) E (x) E (y)- 1E (z) = E (x--y + z). If ~ is any unit in R and x, yeR, then by (2.3), E(x) E (~- I) E(y) _-- -- E(x) E (=)- 1D(=) E (~)- ~ E(y) = -- E (x-- =) E(o) D(~) E(o) E(y-- =) --__ E(x--=)D(~-a)E(y--=), where we have used (e.3), (2.7), (2.4) and (e.5). Replacing ~ by ~-1, we find (2.9) E(x) E(~)E(y) =E(x--~-~)D(=)E(y-- ~-I). Finally if~, ~ are any units, and x,y~R, then E(x)E(~ -~ I)E(~ q- I)E(y) = E(x)E(e)E(o)E(I)2E(o)E(~)E(y) ---- -- E(x)E(~)E(-- I)E(~)E(y) ----- -- E(x-- ~-I)D(~)E( - I -- ~-I-- ~-I)D(~)E(y--~-'), 368 ON THE STRUCTURE OF THE GL~ OF A RING where we have used (2.6), (2.5) and (2.9) in turn. Hence (~z.xo) E(x)E(~ + I)E({3.-t- i)E(y) = -- E(x-- ~-~)D(e)E( - ~--~-l--~-~)D(~)E(y--~-l). We conclude this section by showing that in any ring, De normalizes Ig e and by obtaining a certain standard form for the elements of GE e which will later be shown to be unique, for suitably restricted rings. Proposition (2. x). -- In any ring R, GE2(R ) = DE = ED and E is normal in GE~(R). Proof. --By (2.4), (2.5) and (2.3), [o~, ~]-'E(x)[~, ~] = [,-~, ~-~]E(~-~x~) = D(,-~)E(~-~x~) Thus IgDclg and since GEe(R ) is generated by D and E, we have E<] GEe(R ) and the result follows. We note that in general DnE~= i, so that we do not have a semi-direct product. In fact, by (2.5) and (2.3), D(o~)eDnE for all ,EU(R); later (in w we shall find conditions under which the subgroup generated by the D(e) is exactly DnE. By definition, every element of GEe(R ) is a product of matrices [0~, ~], E(x) and E(x) -1. Now by (2.6) any factor E(x) -1 can be replaced by a product of E's, so it follows from Proposition 2. I that every element A of GEe(R ) has the form (2. x,) A= [~, ~]E(a~)... E(a,). If for some i in i<i<r, ai=o, then we can shorten (2. II) by using (2.2); if aieU(R ) we can use (2.9) and (2.4) to shorten (2. i I). Thus after a finite number of steps we reach a form (2.ii) which cannot be shortened in this way and then a~r ) for i <i<r. Moreover, when r =-- 2, we may assume by (2.5) that al, a2 do not both vanish. Such an expression for A is s:dd to be a standard form. We note that only (2.2-4) and their consequences have been used in obtaining the standard form. Our results may now be summed up as follows: Theorem (2.2). -- Let R be any ring and denote by GEe(R ) the group generated by all matrices (., ~U(R)) and E(a)=(__: Io) (aeR), and set D(o~) = [o~. o~-1]. These generators satisfy the relations: E(x) E(o) E(y) ---- --E(x +y), (2.2) (2.a) E(,z) E(.-~)E(~) = -- D(.), E(x) = (2.4) "2 Io P.M. GOHN These relations, together with those in the group D(R) generated by the [% [~] imply E(o)'=--l, g(~)~=--l, E(--I)3=I, (2.5) (2.6) E(x) -t = E(o)E(--x)E(o), E(x) E (y) -t ---- E (x--y) E (o) -~ -- -- E (x--y) E (o), (2.7) (2.8) E(x)E(y)-tE(z) -= E(x--y + z), E(x)E(~-a)E(y) = E(x--cz)D(a-t)E(y-- e), (~'.9) (2.xo) E(x)E(~+ I)E(~ + i)E(y) = -- E(x--~-X)D(~)E( - i --e-~--~-t)D(~)E(~--~-1). Moreover, they imply that the group E2(R) generated by all E(a) /s normal in GE2(R ) and that eve O, element A of GE2(R ) can be expressed in standard form (2.II) A = [=, NE(al). 9 9 E(a,), where ~, ~zU(R), a~zR and such that a~CU0(R ) for I<i<r and al, a 2 are not both zero in case r = 2. In some rings it is possible to shorten (2. i I) still further by an application of (2. i 0), but for many rings such a reduction is impossible and (2. r I ) actually represents a normal form for the dements of GE~(R) (cf. w167 4 and 7). Such a ring is said to have a unique standard form for GE 2. We shall also meet rings R for which the only relation in GE2(R ) of the form W = I, where W is a word in standard form, is the trivial rela- tion I = I. Such a ring is said to be quasi-free for GE 2. It is clear that of the properties: (i) R has a unique standard form for GE2, (ii) R is quasi-free for GE2, (iii) R is universal for GE2, each implies the next, but as examples to be given later show, these three classes are distinct. 3. Direct and free products of GE-rings. We now turn to consider ring constructions which preserve the property of being a GE-ring. Here it is more convenient not to restrict the size of the matrices in any way. For any ring R and any integer n/> I we may define GEn(R), Dn(R ) and En(R ) as the subgroups of GLn(R) generated by all elementary matrices, all diagonal matrices and all Bq(a)(aeR, i#j) respectively. As in the case n=2, it is easily verified that EnD ~ = D~E n = GEn (R), and hence in every GE-ring, every AzGLn(R) has form (3. x ) A = DB(t)... B (r), where DeD, and B(PI=Bq(a) for some i#j, aER (depending on p). Let R be a direct product of a family of rings, say R=HRx, and denote the canonical projection R~R x by %. Then zx is a homomorphism which induces a 370 ON THE STRUCTURE OF THE GL~ OF A RING Ix group homomorphism ~x : GLn(R)-->GLn(Rx)" Composing these homomorphisms, we obtain a mapping F : GL~(R) ~ IIGLn (Rx) , which is easily seen to be an isomorphism. Now suppose that each R z is a GE-ring; then R need not be a GE-ring. We need only take a product of infinitely many factors R x such that in R z the integer r in (3. i) cannot be taken to be bounded, as X varies (1). Then it is easy to write down a matrix A~GLn(R ) which is not of the form (3. i). However, when the number of factors R x is finite and each is a GE-ring, then any AeGLn(R ) is again of the form (3. i) and we obtain Theorem (3. I). -- The direct product of a finite number of GE-rings is again a GE-ring. The result holds more generally for the direct sum (=weak direct product) of any number of GE-rings, since any matrix over the direct sum lies in the direct product of a finite number of factors. We next suppose that we are dealing with K-algebras, K being a commutative ring with I, and ask whether the GE-property is preserved under tensor products. The answer is easily seen to be " no ", even when K is a field, since K[x] is a GE-ring, for any indeterminate x over K, whereas the polynomial ring in two indeterminates, K[x,y] =K[x] | is not (see w 7). In fact the analogy with the Euclidean algorithm suggests that we consider, not the tensor product but the free product (cf. [5]). Of course it is now more natural to replace the K-algebras by K-rings, where K is any ring with I. It will be recalled that a K-ring is a ring R with a canonical homomorphism 0 : K---~R. In case 0 is injective, R is said to be a strict K-ring: thus a strict K-ring is just a ring in which K is embedded in a canonical way. In considering free products of rings we shall assume that all the factors are strict K-rings for a fixed K, which may thus be considered as a common subring of all the factors. To answer the question whether the GE-property is preserved by free products, would require a formidable calculation; we therefore confine ourselves to establishing the result in a special case (which turns out to be sufficient for many applications). Namely, we shall assume in addition that our rings are locally free ideal rings, i.e. semifirs (2). The GE-rings which are semifirs are characterized in the following Proposition (3-2). -- For any ring R, the following assertions are equivalent: (i) R is a GE-ring and a semifir, (ii) for any n >~ i, given al, ..., an, ba, ..., b,~R, if (3.2) Y~a~b i ~o, bl, ..., bn not all zero, then there exists CEEn(R ) such that (as, ..., an)C has at least one zero coordinate. (x) We shall see bclow (in w 5) that there even exist GEe-rings in which r cannot be taken bounded. (2) Cf. [6], where they are called local firs. This name has now been abandoned as it may give rise to confusion with local rings. Thus a semifir is an integral domain (not necessarily commutative) in which all finitely generated right ideals are free and any two bases of a given free module have the same cardinal. 871 x~ P.M. COHN A ring satisfying either (and hence both) of (i), (ii) will be described as a GE-semifir, for brevity. We note that H. Bass in [3] takes (ii) as the definition of a generalized Euclidean ring. Proof. -- Suppose (i) holds and (3.2) is given, then by Theorem 2.6 of [6] there exists AeGL,(R) such that (al, ..., a,)A has a zero coordinate. Since Ris a GE-ring, A-I=DE, where DED,, EeE n. Hence (al, ..., a,)E-1D -1 has a zero coordinate, and so does (al, ..., a,)E -1. Thus R satisfies (ii). Conversely, assume (ii) ; then R is a semifir, again by Theorem 2.6 of [6], so it remains to prove that GE, holds and here we may clearly assume that n> I. Let A be an invertible n � n matrix; if the first row has only one non-zero element then this must be a unit. By permuting the columns of A (which corresponds to right multipli- cation by a matrix in E,) we may bring this unit to the (I,I)-position and obtain :C o o o) * A 1 where AleGL,_I(R). Now the result follows by induction on n. If A has preci- sely r(> i) non-zero elements in the first row, we may assume that these elements come in the first r places. Writing A=(a0), A-l=(~ij.), we have, for j=2, ..., n, (S-S) i Zla. ~i = o. Since A -1 is invertible, either ~1i4:o for some j~>2 or a~i:~o for some j~>2. Choosing j accordingly, we may assume that of the % actually occurring in (3.3), not all vanish. By (ii) there exists BleEr(R ) such that (an, ..., al,)B 1 has a zero coordinate. Hence A(o has fewer than r non-zero elements in the first row and after at most n--i steps we reach the case r = I treated before. This completes the proof. In connexion with this proposition we remark that a GE-ring need not be a semifir. For by Theorem 3. I, a GE-ring need not even be an integral domain. To obtain an example of a GE-ring which is an integral domain but not a semifir, take any commu- tative local domain (i.e. an integral domain with a single maximal ideal) which is not a valuation ring. This is not a Bezout ring and hence not a semifir, but as we shall see later, any local ring is a GE-ring (w 4 below; see also [I2]). In the other direction we note that a semifir need not be a GE-ring; an example of a commutative principal ideal domain which is not a GE-ring will be given in w 6. We now show that the free product of any family of GE-semifirs over a skew field, 372 ON THE STRUCTURE OF THE GL 2 OF A RING is again a GE-semifir. Let R be any strict K-ring; by a suitable identification we may regard K as a subring of R. If K is a direct summand of R, as right K-module, R = K| then R is called an augmented K-ring, with augmentation module N. It was shown in [6] that for any ring K, the free product of any family of augmented K-rings exists and is again an augmented K-ring. We shall use the terminology and notation for free products introduced in [6]; in particular (H ") is the usual filtration of the free product (by height) and two elements a, b of the free product are said to interactif h(ab) <h(a) -}- h(b). In any ring R, two n-tuples (al, ..., an) and (ba, ..., b,) are said to be GE-equivalent if there is a matrix P=(p~i)eGE,(R) such that bj=Y_,aipli. Lemma (3.3). -- Let K be a field (possibly skew) and (Rx) a family of GE-semifirs. Denote the free product of the Rx (over K) by P and let (H") be the filtration by height of P. Let % b k (k = i, ..., r) be any elements of P such that h(akbk)-~ n and (3.4) ~akbk----o (mod H~-I). Further, assume that the a k are ordered by decreasing height, say h(ai)=m for i<<.s and h(aj)<m for j> s. Then either a) (al, ..., as) is GE-equivalent over K to an s-tuple (a't, ..., a's) such that h(a~)<~m with strict inequality for at least one i, or b) the elements a~ (i<~s) which interact with b i in a given factor R x are GE-equivalent over Rx to a tuple of elements of height <<. m, with strict inequality in at least one place, or c) for each a~ (i<<.s) there exist elements xqeP such that h(a~--Zaixi~)<m , h(a~xj~)<~m (j>s). The proof of this lemma will be omitted as it is very similar to that of Lemma 4- I of [6], the only difference being that instead of unimodular equivalence we assume and prove GE-equivalence throughout (1). As in [6], Theorem 4.2 we now deduce Theorem (3.4). -- Let K be a field (possibly skew) and (R~) a family of GE-semifirs. Then the free product of the Rz over K is again a GE-semifir. In particular, all the examples of semifirs given in [6] are actually GE-semifirs. For all arise as free products whose factors are either fields or rings of the form k[t] or k[t, t -1] and these are all clearly GE-rings. In particular, it follows that the group algebra over k of a free group (and the semigroup algebra of a free semigroup) are GE-semifirs, a fact used by H. Bass in [3]. (1) Note that the phrase " set including an element of height less than m " occurring twice in Lemma 4. x of [6] should each time be replaced by: " tuple of elements of height ~<rn, with strict inequality in at least one place ". Further, K should be a skew field. In fact this is what was proved (and used) there. 373 14 P.M. COHN 4- GL2(R) for a local ring. It is well known that any field (not necessarily commutative) is a GE-ring (cf. e.g. Dieudonn6 [8]) and this result was generalized to local rings by Klingenberg [I2]. In the present section we shall prove that a local ring is in fact a universal GE2-ring; however it does not have a unique standard form and in fact is not even quasi-free for GE~ (unless it is a field). To investigate the uniqueness of the form (4.I) [ct, ~]E(al)...E(a,)=I (~, ~eU(R), a#U0(R ) I<i<r), for any ring R, let us look at low values of r. Clearly, when r= o, we must have e=~=I. The case r=I is impossible, as we see by comparing (I,2)-elements. When r=2, we have eal=~a2=o, hence a l=a~=o and c~=~=--i. For r=3 a comparison of (2,2)-elements gives ~a~ = i, hence a 2 is a unit, which contradicts the conditions in (4. I). Thus in any ring R the relation (4. I) (with the stated conditions) is possible only if r 1> 4, apart from trivial cases. Next let r= 4 and for convenience write the relation as E(a~)F~(a~)F~(a3)E(a4) = [~, ~]. A comparison of terms shows that this holds if and only if ala~aaa4--alas--a3a4--ala 4 + I = o~, a 1 a s a a- a 1- a 3 -~ o, a s a 3 a 4- a s- a 4 = o, a 2 a 3 -- I = -- ~. We shall now show that in any local ring R which is not a field, these relations can be satisfied by elements ~, ~U(R), as, a3r ). For by hypothesis, U0(R ) 4= R; choose any elements a2, a 3 not in U0(R), then ~ = I--a~a3eU(R ) and al, a 4 are given by the equations al~+a3=o, ~a4+a2=o, which determine al, a 4 uniquely as elements of R; moreover it is clear that al, a 4 so defined do not lie in U0(R ). The fourth relation is of the form e ----- I + nonunit, and it determines e~U(R). Thus for r=4, (4. I) can be satisfied in any local ring not a field, hence such a ring is not quasi-free for GE~. Let R be a ring in which any one-sided inverse is two-sided and which satisfies the following condition: for any n> I, given al, ..., a~eR such that ~aiR= R, there exist b~, ..., b~_~R such that Y~(a~+a~bi)R=R. This is expressed by saying that n = I is a stable range for R; any such ring is easily seen to be a GE-ring (cf. H. Bass [2], who proves this and also shows that these rings include any ring which modulo itsJacobson radical satisfies the minimum condition on left ideals). The case n=2 of the above 874 ON THE STRUGTURt~ OF THE GL a OF A RING x5 condition states that for any a, beR such that aR+bR--R, there exists ceR such that a + beaU(R) ; clearly this holds in any local ring. Assuming this condition we can easily show that any AeGL2(R ) can be written in the form (2. i i) with r~< 3. For by hypothesis, for any B~GL2(R ) we can find ueR such that BE(u) -1 has a unit in the (I,2)-place and hence, for suitable wR, BE(u)-IE(v) -1 has 0 in the (I,2)-place. This matrix is therefore of the form Applying this argument with E(o)A in place of B, we find that E(o)A------ [e, ~]E(o)E(~-Ic)E(v)E(u), i.e. A =-- [~, e]E(~-%)E(v)E(u), which is of the required form. This shows in particular that any local ring is a GE2-ring. We now come to Theorem (4-x). -- Any local ring is a universal GE~-ring. After what has been said it only remains to show that any relation (4. i) in a local ring is a consequence of tlhe defining relations (2.2-4) and the relations in D,. Let (4.2) [oc, ~]E(al)... E(ar):I , ~, ~U(R), aiCUo(R ) for I<i<r. If i~>4, then as, a3r and hence as--I , a3--I~U(R), say a2---=y-[- i, a3=~+ I. By (2. Io), we can reduce (4.2) to --[~, ~]E(al--y-t)D(y)E(-- ~ --y-t--~-l)D(~)E(a4--~-t)E(as)... E(a,)= ~. Using (2.4) to pull D(y) and D(~) through to the left and then (2.9) (as in the proof of Theorem 2.2) we obtain an expression of the form (4.2) but with r replaced by a smaller value. This process can be continued as long as r/> 4; so we finally reduce (4.2) to the case r~< 3 and only (2.2-4) and their consequences have been used in the reduction. For r~< 3 we saw that the only relation (4.2) is --E(o)2=I and by (2.5) this is also a consequence of (2.2-4). Hence (4.2) itself is a consequence of (2.2-4) (and relations in D), as we wished to show. 5- Discretely normed rings. We now look for more general conditions on our ring R to ensure that the form (2.ii) for the elements of GE2(R), given by Proposition 2.2 is unique. The most natural way of imposing such conditions is by means of a norm; this is of course closely related to the question of the existence of a Euclidean algorithm, or a weak algorithm (cf. [5]) in R and hence to the question whether R is a Euclidean ring. In fact we shall use this method to construct rings which are not Euclidean in a rather strong sense: they are not GE-rings. 375 16 P.M. CIOHN Definition. -- A norm on R is a mapping I } from R to the nonnegative real numbers such that N. I. Ix[=o /fand0nly if x=o, N. 2. [x+y[<<.lxl+ly ], N. 3. lxy[ =lxl [Yl. We note that by N. 3, R must be an integral domain (not necessarily commutative). We shall say that I I is a discrete norm or that R is discretely normed, if further, N. 4. ]x]/> r for all x+o, with equality only if xeU(R), N. 5. there exists no xeR such that I<]xl<2. If 0~ is any unit, then by N. 4, ]e[>l ~, ]~162 ~> i, while by N. 3, hence I~[=I for all ~eU(R). Now N. 5 shows that [xl>~2 for any xCU0(R). Conversely, the two conditions N- 4' l~f=~ for an ~UCR), N. 5' I x]>12 for all xCU0(R ) imply N. 4 and N. 5- In order to derive further consequences of N. I- 5 we need an expression for the product E(al)... E(a,); this is closely related to the chain of equations in the Euclidean algorithm, cf. w 8. Let tt, t~, ... be any noncommuting indeterminates and define a sequence of polynomials in the t's with integer coefficients recursively by the equations e 1=0 , no= I~ (5"x) [e,(q, ..., t,)=e,_l(tl, ..., t,_l)t,--e,_z(tx, ..., t,_2). We note that for n i> o, the suffix of e, just indicates the number of arguments and so may be omitted when the arguments are given explicitly; we shall do so in what follows and only write the suffix when the arguments are omitted. We assert that _(e(aD...,a,) e(a~, . .., a,_l) ] (5.2) ( ~ " ., a,) --e(a2, ., at_l) ]" This is clear for r~ I and in the general case follows by induction, since, writing e~=e(at, ..., a~), e~ =-e(az, ..., a~+l) , we have ~--- r n 9 --e,_l From the symmetry of (5.2) it is easy to see that the e's may also be defined by e_t=o , no=I, e,(q, ..., t,~)= tle,_l(t 2, ..., t,)--e,_~(t3, ..., t,). The following lemma is basic for all uniqueness questions in GE2(R ) where R is discretely normed. E'aa)'"E'a"--\--e(a2,( ON THE STRUCTURE OF THE GL, OF A RING 17 Lemma (5- x). -- Let at, ..., a, (r> I) be any elements of a discretely normed ring R such that ai$Uo(R ) for i>I. Then (5.3) {e(ai, ..., ar)] >t {e(al, ..., a,_,) I and (5.4) e(al, ..., at) =~ O. If, moreover, the norm on R is such that (5.5) for any u eR, [ u ] = i and {I ~-U [ = 2 imply u =i, then the inequality in (5.3) is strict, unless 121 =2 and at=~eU(R), a2=a 4 ..... 2~ -1 and a3 = an ..... 2e. In this case, E(al)... E(a~)equals (: :) or (: :> according as r is even or odd. Then for i> I, {a~{ 1> 2, hence Proof. -- Write e~ = e(a t, ..., a~) for short. lei[ = {ei_tai--e~_2[ >1 2 {ei_t[-- [e~_2[, i.e. (5.6) le, l--le~-t[ ~> [e,-tl--{e~-2l (i> i). Suppose first that at=o; then el=al=o , e2=ata~--I =--I and hence by (5.6), ] erl--ler_l] i> le2l-lell = I. This proves (5.3) (with strict inequality) and hence (5.4)- Now let al+ o, then {a 11 >/ I and (5.7) [e~]--{eo] = [a~l-- I >~ o. Combining this with (5.6) we see that o for i>i, and hence (5.8) [er] >/le,_~{ t> .../> ]eli t> I. This proves (5-3) and (5-4). If further, lal[> i, then we have strict inequality in (5-7) and hence in (5-3). Likewise, if }al}> 2 for some i> i, then we have strict inequality in the corresponding formula (5.6) and hence in (5.3). There remains the case [at{ = I, {a~] = 2 (i> I); in particular, a~ is then a unit. Suppose first that at= I and assume that (5-5) holds. If we have equality in (5.3), then [ala~--I{=I, so ala2=i+b, where [b[=I, lI+b[=[ala2[=2 , hence b=I anda2=ataz=2. Now In general, when a x = and by induction it follows that all the al (i> i) must be 2. is any unit, we have I]A= [~-', I]E(~)E(a2)... E(ar) = E(I)E(a2~)E(~-la3)E(a,~)... 3 18 P. M. GOHN with a factor D = [I, g-l] (for r odd) or [~-t, I] (for r even) at the right-hand end, as follows by repeated application of (2.4)- Now if the elements of the first row of A have the same norm, then the same holds for [~-1, I]AD-1, so we are reduced to the case at= I and the result follows. Theorem (5.2). ~ Any discretely normed ring is quasi-free for GE2, and hence is universal for GE2. Proof. -- By Theorem 2.2, any relation in GE2(R ) can be brought to standard form [=, ~]E(at)... E(a,)= I, (5.9) for i <i<r. Comparing the (i,2)-elements in this equation we see where a~Oo(R ) that e(al, . .., at-l)=% and this contradicts Lemma 5- i if r-- I > i, i.e. r> 2. But when r~< 2, only the trivial cases noted in w 4 are possible. This completes the proof. Corollary. ~ A discretely normed ring which is a GE2-r/ng is a universal GE2-ring. When R is a discretely normed ring, it is easy to obtain an explicit form for the involutions in GE2(R), i.e. the elements of order two. This will be useful later on, in the study of automorphisms of GE~(R) and also in showing that for certain Dedekind rings the class number must be I. We begin by determining the centre of GE2(R), where R may be any ring: Proposition (5.3). -- For any ring R(4: o) the centralizer of E2(R) in GL2(R) consists of all matrices XI, where X runs over the central units of R. Proof.-- Let A=( a ~) ~GL2(R) centralize E2(R) ; since A commutes with E(o) we must have a=d, b +c=o; since it commutes with E(I) we have b =% i.e. A=aI and now the commutativity with E(x) (x~R) shows that a lies in the centre of R. This completes the proof. We next determine how the standard form simplifies in the case of elements of finite order. Proposition (5.4). -- Let R be a ring which is quasi-free for GE 2; then any matrix in GE2(R) which is of finite order modulo the centre of GE2(R) is conjugate (under E2(R)) to an element of the form (5. or (5" II) [~, [3]E(o)E(b), and in case (5. IO) holds, we have acUo(R ). 878 ON THE STRUCTURE OF THE GL 2 OF A RING x 9 Proof. -- Let A~GE2(R ) be of finite order modulo the centre of GE2(R ) and write (5. x2) A = [~, 13]E(a~)... E(a~). Then any conjugate of A is again of finite order mod the centre. If r>~ 3 in (5. I2) and alEU0(R), then A is conjugate to [9, ~]E(a')E(al)... E(a,-1), for some a'eR, and now the length can be reduced. Likewise, if areU0(R), we can transfer the factor E(al) to the right-hand end and then make a reduction. Thus we obtain a form (5-12) for a conjugate of A, in which either r~<2 or a#U0(R ) for all i. Since A'*=XI, we have [~, ~]E(al)... E(ar) [a, ~]E(al) ......... E(ar) =XI. By pulling all the diagonal factors through to the left and then transferring them to the right-hand side, we obtain a relation E(bl)... E(b,n ) = diagonal matrix, where each b i is associated to some a~. By hypothesis this can only hold if some bfiU0(R), hence some a~eU0(R ) and so in particular, r~<2. For r= I, we obtain the form (5. io), where neces,~arily aleU0(R ). If r----2, we may without loss of gene- rality assume that aleU0(R). Suppose first that al=TeU(R). Then transforming A by E(x), we have E(x)AE(x) -1 =E(x)[~, ~]E(y)E(b)E(o)E(--x)E(o) =--[~, ~]E(~-lx~--y-1)D(y)E(b--y-l--x)E(o). Choose x=b--y -~, then the right-hand side becomes and this is again of the form (5. I o). In the alternative case, a 1 = o and we obtain (5. i i). Finally if in (5.12) r=o, then this is of the form (5.II), with b=o. We remark that (5-IO) is conjugate, under GE~(R), to a matrix of the form [I, ~]E(a) ; for we can reduce c~ to I by transforming by [0r I]. It is now easy to derive a standard form for the involutions in GE2(R), when R is quasi-free for GE 2. Theorem (5.5). -- Let R be an integral domain which is quasi-free for GE2, then any invo- lution ~:--I in GE~(R) is conjugate to one of the form where h runs over a transversal of 2R in R (qua additive group). In particular, if 2R=R, then every non-central involution is conjugate to [i,- I]. 379 20 P.M. GOHN Proof. -- By Prop. 5-4 and the remark following it, any involution A+--I is conjugate to one of the forms (_; ;) In the first case, by squaring, we see that [~, ~]E(~-la)E(a)=I, hence a=o, ~=--i and we find A=[I, --I]E(o). Transforming by [I, --I]E(I)E(o) -1 we reach (5-I3) with h=I. In the second case squaring shows that ~2=~=I, hence e,~=4-i (because R is an integral domain) and ~(u + ~)b = o. Thus either -- ~ = e = 4- i and we have the form (5.i3) or ~=~=4-i and b=o. But this means that A= and these possibilities were excluded. Thus every involution can be brought to the form (5. I3). Now transforming (5. I3) by B~l(c ) we obtain 4- I and by a suitable choice of c we can ensure that h 1 = h-- 2 c belongs to the given trans- versal of 2R in R. In particular when 2 R = R this means that every involution is conjugate to :k [i, -- i]; since [-- i, I] is transformed to [I, -- I] by E(o), we need only one of these forms and the proof is complete. A ring is said to be 2-torsion free if 2 x = o implies x = o. With this definition we note the following application: Proposition (5.6). -- Let R be a 2-torsion free integral domain which is a quasi-free GE~-ring. Then every projective module on two generators is free. Proof. -- Any projective module M on two generators is a direct summand of R 2 and may be characterized by its projection, an idempotent 2 � 2 matrix, E say. The matrix P=I--2E has square I; if P=4-I, then E=o orI, andMis clearly free. Otherwise P is an involution *--I and by Theorem 5.5, after applying a suitable inner automorphism, we have o) In particular h -- o (mod 2 R), hence by a further inner automorphism we may arrange that P=4-[I,--I]; then E=[i,o] or [o, I] and M is again free. In a Dedekind domain each ideal is projective and can be generated by two elements; moreover, the free ideals are just the principal ideals, hence we obtain the Corollary. -- Let R be a 2-torsion free discretely normed Dedekind ring. If R is a GE2-ring, then it must be a principal ideal domain. In the next section we shall see how the result may be applied to purely imaginary quadratic number fields. We conclude this section by giving some examples of discretely normed rings. i. The rational integers, with the usual absolute value. More generally, consider the ring of algebraic integers in any imaginary quadratic number field. If this is 380 ON THE STRUCTURE OF THE GL 2 OF A RING 2! embedded in the complex numbers, the usual absolute value is a norm, i.e. it satisfies N. 1- 3. N. 4 is also satisfied because the integers form a lattice which can only have finitely many points in the unit circle; if any were strictly inside the unit circle, their powers would give an infinite set. Further, if an element is on the unit circle, so is its conjugate and hence the: element is a unit. Thus N. 4 always holds; N. 5 holds with a few exceptions (see w 6). Whenever it is satisfied we have a discretely normed ring. 2. Let R be any ring such that U0(R) is a field, k say, and suppose that we have a degree-function on R, i.e. to each a sR there corresponds a non-negative integer or --0% written d(a), such that D. I. d(a)=--oo if and only if a=o, D. 2. d(a)=o if and only if aEU(R), D. 3. d(a--b)<~max{d(a), d(b)}, D. 4. d(ab) -= d(a) § d(b). Then R becomes a discretely normed ring if we put l al = 2 a(~ For example, the polynomial rings over k, in any number of indeterminates, and the free associative algebras (polynomial rings in non-commuting indeterminates) are of this form. 3. If R is any discretely normed ring, then R[x], the polynomial ring in a single indeterminate over R, may be discretely normed by the rule l Y'a,x'l = Y la, i 2'. 4. The free product of fields does not at first sight admit a norm, but it does have a filtration which is very nearly a norm, and many of the results proved here for discretely normed rings can be carried over for free products of fields. We shall not enter into the details here (cf. [5])- In a discretely normed ring it is often possible to decide whether the ring is a GE2-ring. If it happens to be Euclidean with respect to the norm, it is clearly a GE~-ring. In the next two sections we shall show that under certain conditions a discretely normed ring cannot be a GE2-ring; this means in particular that no norm function can be defined for which the ring is Euclidean. 6. Rings of algebraic integers and algebraic functions. We have just seen that the ring of integers in an imaginary quadratic number field is discretely normed, provided that N. 5 holds. This amounts to the condition that no integer a in the ring satisfies I<N(a)<4. It is easily verified that this holds for the integers in Q(~v/~----d), where d is positive and squarefree, except when (6.x) d=I, 2, 3, 7, II. 381 22 P.M. COHN As it happens, these are just the fields in which there is a Euclidean algorithm with respect to the usual norm [a[=N(a) 112 (cf. [9], ch. 14). For the remaining cases we have Theorem (6. x), -- Let d be a square free positive integer and I the ring of integers in Q (~-~--d). Then I is not a GE~-ring, unless d has one of the values (6. I). Proof. -- We shall assume that I is a GE2-ring for some d which is not among the values (6. I) and derive a contradiction. By what has been said, I is discretely normed by [a[=N(a) 1/2, and since I is also a 2-torsion free Dedekind ring, it is a principal ideal domain, by Proposition 5- 6 Corollary. By Theorem 2.2, every element AeGL2(I ) has the form (6.2) A=[~, [~]E(qt)...E(q,), where [q~]1> 2 for i<i<r. Since [ ] is the ordinary distance in the Euclidean plane, condition (5- 5) of Lemma 5- i is always satisfied. Moreover, the only units in I are -4- I ; this is easily checked because the low values (6. i) of d have been excluded. Let us denote the first row of A by (a, b) ; then three cases are possible, according to the value of qr. (i) [q,[ 1> 2. Then by Lemma 5. I, [aI>[b[, unless a = (recall that Q-I are the only units in I). (ii) ]qr]=o and hence q,=o. Applying Lemma 5.I to AE(o)-!, we see that [a[<] b 1, unless a =+b. (iii) ]qr] = I and hence qr =-4-I. In this case we find that (6.3) ]a~bl<lbl for at least one choice of sign, unless a-4-b=b. Again this follows by considering AE(q,) -~. Now suppose that [a]=[bl, but a:#-4-b. Then the first two alternatives do not apply and a:~o, 2b, so that (6.3) must hold for at least one choice of sign. Thus if we can find a and b, forming the first row of an invertible matrix, such that (6.4) lal=lbl, [a+bl>~lbl, we have a contradiction. Now in a principal ideal domain any pair of elements without a common factor forms the first row of an invertible matrix, and any element of the field of fractions can be written as a quotient of two elements without a common factor. So to satisfy (6.4) we only have to find an element 0~EQ(%/~) such that (6.5) lOCI=I, [0r I; for on writing 0~ = a/b in reduced form, we obtain a solution of (6.4). Now the equa- tion [ 0~] = i defines a circle in R ~ (relative to oblique coordinates) on which the rational points are clearly dense. These correspond to points c~eQ(~/~) and by going suffi- ciently far from the real axis we can ensure that the second relation in (6.5) is also satisfied (for both signs). This gives us the required pair of values (a, b) and it shows that I is not a GE2-ring. 882 ON THE STRUCTURE OF THE GL~ OF A RING The first value of d not listed in (6. i) for which I is a principal ideal domain is d=i 9 (cf. [9], P. 213) and in this case an explicit matrix in GL2(I ) but not in GE~(I) is 2+o 3--0 5--20]' where 03--0-{-5 =o. By going through the steps of the proof of Proposition 5.6, Corollary, we can construct a similar example for Q(~/'~-5). In the language of valuation theory we can say that for a ring of integers in an algebraic number field to be discretely normed it is necessary for the field to have a single place at infinity. When this condition holds we can prove an analogue of Theorem 6. I for function fields (1). Theorem (6.2). -- Let K be afield of functions of a single variable with k as fMd of constants and assume that k has characteristic not two. Let p be any place of K/k of degree fp> I and denote by I the ring of elements of K which are integral everywhere except possibly at p. Then I is not a GE2-ring. Proof. -- If v~ denotes the exponential valuation at p then the product formula shows that vp(x)~<o for all xEI, x~eo, with equality only if x~k. If we define ~(x) =--v~ (x), then ~ is easily verified to be a degree function on I. Hence I is discretely normed and since it is 2-torsion free and Dedekind, we see that if I is a GE2-ring, it must be a principal ideal domain. Further, every AEGL2(I ) has the form A=(ca ~:) = [~, ~]E(ql)... E(q,) ~(q,)>o (z<i<r). As in the proof of Theorem 6. z there are three possibilities: (i) 3(fr)>o: then ~(a)>~(b) oi a=Xb (X~k), (ii) qr=o: then 3(a)<8(b) or a=Xb (Xek), (iii) 3(q,)=o: then q,;=Xek and S(a--Xb)<S(b). To avoid these three possibilities we need only find a pair of elements a, b in I without common factor such that S(a--Xb)=~(b) for all X~k. Let k(p) be the residue field at p and x-+x the residue mapping. By hypothesis, k(p)~e k, so there exists ~ek(p), ~r choosey~K suchthat ~=~. Then ~(y--X)=o forall X~k and as in the proof of Theorem 6. I we can write y = a/b, whence ~(a--Xb) = 3(b), so that none of (i)-(iii) can hold. Hence I cannot be a GE2-ring. When K is of genus zero, the condition of Theorem 6.2 is necessary and sufficient, for in that case .f~ = I implies that K is rational and I is just a polynomial ring over k. (1) I am indebted to J. V. Armitage for a helpful discussion of this point. 883 P. M. COHN ~4 7. k-rlngs with a degree function. We now turn to look at the second example of discretely normed rings given in w 5- Let k be a field (possibly skew), then a k-ring with a degree function was defined in w 5; we stress that an element of degree zero must by definition lie in k. In w 5 we showed that such a ring is discretely normed. We shall now show that for these rings we can strengthen Theorem 5.2 by proving that the standard form is unique. Theorem (7- I). -- Let R be a k-ring with a degree function (k afield); then R has a unique standard form for GE 2 . Proof. -- By Theorem 2.2 every AeGE2(R ) can be brought to the form (7. I ) A= [~, ~]E(al)... E(ar) (ai~R , o~, ~@U(R)), where a iek (I <i<r) and al, a s are not both zero in case r = 2. Suppose that we also have (7.2) A=[y, ~]E(bl)... E(b,) bi6k (I<j<s); we must show that ~=y,~=3 and r=s,a~:b~. We may assume r~>s and then use induction on r; further, we may assume that -(=~= I, by replacing A by [y, ~]-IA if necessary. If a, = b~ we can cancel the last factor and use induction, so we may assume that a,+b,. Then I--+[,, ~]E(al)... E(ar)E(o)E(--bs)... E(--bl)E(o ) = :t:[~, ~]E(al)... E(ar--b,)E(--b~_~)... E(--ba)E(o ). If we put a,--b~=c and multiply by E(o) we get (7.3) E(o) =DE(a1)... E(a,_~)E(c)E(--b~_a)... E(--bl) , where D is some diagonal matrix. Comparing elements in the (~,2)-place we find (7"4) e(a2, ..., at_l, c,--b,_l, ...,--b~)=o. We now apply Lemma 5. I and note that condition (5.5) is satisfied because for R the hypothesis is vacuous, and [ 21~< I ; therefore the inequality in (5.3) is always strict. Assume first that s~>3, then also r~> 3 and we reach a contradiction unless cek. But then c must be a unit because it is not zero, and (7-3) takes the form E(o) =DE(a~)... E(a,_l--c-1)D(c)E(--bs_t--c-~)E(--b~_2)... E(--bl). By hypothesis, at_a, b,_lek, hence ar_l--c -1, b,_l--c-lek. If we pull D(c) through to the left (by (2.4)) and equate the (2,2)-elements, we obtain e(a;, ..., a;_2, a'~_~--c, --b~_l--c -1, --b,_ 2, ...,--be)----o, where a~ is associated to a i and hence none of the arguments lie in k; but this contradicts Lemma 5- I. 384 ON THE STRUCTURE OF THE GL, OF A RING When s=2 we have bib2-- I bI1)=[~, ~]E(al). . . E(at), --b~ hence e(a2, ..., a,_l) .... ~-1. By Lemma 5. i, r-- 2 = O and in this case the uniqueness follows easily. When s = i, E(b) = [~, ~]E(al)... E(a,), hence e(a2,...,ar_l)=o. ByLemma5.i , r--2=--I, so r=i and the uniqueness is then clear. The same argument applies when s----o, and this completes the proof. We remark that the result does not hold for all discretely normed rings. E.g., for the ring Z of rational integers, In the next section we shall introduce a different generating set for GL2(Z ) in terms of which a unique normal form is possible. We now give some more specific examples of rings with a degree-function which are GE-rings and some which are not. Naturally, any Euclidean ring (with respect to its degree function) is a GE-ring, and for this to hold the ring need not be commutative, e.g. the ring k [x] of polynomials in a single indeterminate over a skew field is Euclidean. More generally, we have the rings with a weak algorithm described in [5]- We shall not recall the definition here but merely note that they are always k-rings with a degree function, for some field k. A typical example is the tensor ring over an arbitrary k-bimodule; in particular, any free associative algebra over a commutative field is of this form. For these rings we have Theorem (7.2). -- Any ring with a weak algorithm is a GE-ring. Pro@ -- By Prop. 3.2 we need only show: for any n~>I, given al, ..., a,, bl, ..., b, ~R such that (7.5) ~ga~b~=o b~ not all zero, then there exists CEEn(R ) such that (al, ..., a,)C has at least one zero coordinate. The proof is by induction on }]d(ai). By suitably renumbering the a's and b's we may assume that d(aibi)=m for i<~s and d(aibi)<m for j>s. Then (7.5)shows that al,..., a s are right R-dependent, hence by the weak algorithm, an a s of maximal degree, say a 1 is right R-dependent on the rest: a 1 = ~aic i -~ a; d(a;)< d(al) , d(ai61)(d(al). This shows that the transformation from (al, ..., a,,) to (a'~, a2, ..., an) is a product of elementary transformations and it diminishes the value of the sum ]~d(ai). Hence the result follows by induction. 385 26 P.M. GOHN Since a ring with a weak algorithm is discretely normed, such a ring is a universal GE~-ring; it is also easily seen that E,(R) is normal in GLn(R ). Next, to give some examples of rings which are not GE-rings, we establish a neces- sary condition for a k-ring with a degree function to be a GE2-ring. Let R be any k-ring with a degree function which is also a GE2-ring. We saw that in this case (5.5) always holds and 121~ I, therefore Lemma 5-I shows that in the present case d(e(aD ..., ar) ) > d(e(al, ..., at_l)) for any elements al,..., a, (r>i) such that a~(~k for i>I. Let us call a pair of elements (a, b) a regular row if it can occur as the first row of an invertible matrix over R (when R is commutative this reduces to the notion of a unimodular row defined by Bass in [2]). Take any regular row (a, b) and let A~GL2(R ) be a matrix in which it occurs as first row. Since R is a GE2-ring, A can be written in the form (7. i) and as in the proof of Theorem 6.2, there are there possibilities: (i) if q,r then d(a)>d(b), (ii) if q,=o then d(a)<d(b), (iii) if q~=~eU(R), then d(bo:--a)<d(b). In terms of the notion of (right) R-dependence, which is defined in any ring with a degree function (cf. [4] or [5]), we can sum up the result as follows: Proposition (7-3)- -- If R is a k-ring with a degree function which is also a GE2-ring , then of any two elements of the same degree which form a regular row, each is R-dependent on the other. With the help of this result it is easy to show e.g. that the ring k[x,y] of polynomials in two indeterminates over a field is not a GE2-ring. As the degree function we take the total degree in x andy. Then (I -t- xy, x ~) is a regular row consisting of two elements of the same degree, neither of which is R-dependent on the other. This means that the matrix ( i -~ xy x ~ ) __y2 i --xy which is clearly invertible, cannot be expressed as a product of elementary matrices. The same reasoning shows that more generally, the ring k [Xl, ..., xe] is not a GE2-ring whenever d>I. 8. Discretely ordered rings. The results of w167 5-7 show the usefulness of a norm for the study of GE2(R ). In some respects the same purpose is served by assuming that the ring is totally ordered (1); here it is again necessary to make some discreteness assumption. Definition. -- A discretely ordered ring is a ring R which is totally ordered, such that (8.x) for any a~R, if a>o, then a>~i. (1) I am indebted to H. Bass for drawing my attention to this possibility. 386 ON THE STRUCTURE OF THE GL 2 OF A RING In order to describe GE2(R ) in this case it is advantageous to supplement the diagonal matrices by matrices with positive coefficients and we therefore replace the generators E(x) by (8.2) P(x)=( x Io). Of course the P's could also have been used to develop the theory for discretely normed rings; but the E's there offered the advantage that their determinant was i. If P2(R) denotes the group generated by all the P's, then in particular, [I,--I]=P(I)P(--I)P(I)eP~(R), and since [I,--I] normalizes E2(R) and (8.3) V(x) = [I,- I]E(x), E(x) = [I,- I]P(x), it follows that E2(R) is a subgroup of index 2 in P2(R). In order to describe P2nD 2 we shall write (8.4) C (~) = [., --.- 1] (~ EU(R)). With these notations we have the following analogues of (2.2-4): (8.5) P(x +y) = V(x)P(o)P(y), (8.6) P(.)P(--~-I)P(.) = C(e), (8.7) P(x) [~, ~] = [~, .]P(~-lx.). Now the relation (8.3) between P(x) and E(x) shows that the defining rela- tions (8.5-7) are equivalent to the relations (~. 2-4). Hence R is universal for GE 2 if and only ifGE2 has (8.5-7) and the relations in D, as a complete set of defining relations. As in w 2 we obtain the following relations as a consequence of (8.5"7) : (8.8) P(o)' =I, [P(I)P(-- i)]3=--I, (8.9) V(x)-I = P(o)P(--x)V(o)---- ( ~ _;), (8. zo) P(x) V(y) -1- P(x--y) P(o), (8. z, ) P(x)P(y)-IP(z) = P(x--y + z), (8. x2) P(x) V(e) P(y) ~-P(x -5 .-i)C(-)P(y + e-~). In particular, putting ~ = I in the last relation, we find P(x)V(z)P(y) ---- P(x + i)C(I)P(y + I) ---- C(-- I)P(--x-- i)V(y + i), whence on replacing x, y by x--i, y--z respectively, we find P(--x)P(y) = (3(-- I)V(x-- I)V(1)P(y-- I). (8. ,3) Similarly, P(x)P(--y)P(z) =--P(x-- I)P(I)P(y-- 2)P(I)P(z-- I). (8.z4) Next we shall derive expressions for a product of P's, analogous to (5.2); they 387 28 P. M. COHN are essentially the continuant polynomials, cf. [i8] and [5]. Let tl, t2,.., be sequence of noneommuting indeterminates and define the polynomials p~ recursively b P_l=o, P0=I, (8. x5) p~(tl, ..., tn) =pn_l(tx, ..., t,_t)t ~ +p~_~(tl, ..., t,_~). As in the case of the e's defined in (5- r) we shall omit either the arguments or the suffix, from the p's. It is easily seen that Pla, } - "'" at) p(al, ...,ar_l)] P(al)... \p(a2, ., a,) p(a2, ., a~_l) ]" The symmetry of (8. i6) leads to the following alternative definition for the p's: (8. /P-l=o, P0= , t P,(tl, ..., t,)= t~p,_~(t2, ..., t,)+p,_2(ta, ..., t,). Either definition shows that p, may be described as the sum of t lt2 . . . t n and all tern obtained by omitting one or more pairs of adjacent factors t~t~+t. In particular, th shows the truth of Lemma (8.x.) -- In any ordered ring R, given al, ..., arER such that a~>o J I<<.i<~r (r>o), then p(al, ..., a~)>o. The same conclusion holds if al >~o , and ai>o for 2<<.i<<.r, provided that r >~ 2. This lemma leads to the following analogue of Theorem ~. 2 and Theorem 5- which is related to the uniqueness of the expansion of a rational number in a simp continued fraction (of. [9], P- I35)- Theorem (8.2.) -- Let R be any discretely ordered ring. Then R is universal for GE moreover, any A~GE2(R ) is unique of the form (8. x8) A = [,, ~]P(aa)... V(a,) a~eR, ,, ~eU(R subject to the conditions al/>O , ai~o (I<i<~ (8.x9) (8.20) when r = 2, a 1, a2 are not both zero. Proof. -- By definition, GE2(R ) is generated by the diagonal matrices and all P(c a~R; using (8.9) and (8.7) we can bring any such product to the form (8.i8). ai=o for some i (I<i<r) we can use (8.5) to simplify the expression (8.I8); so may assume that ai4=o (I<i<r). To complete the proof we use induction on t] number of sign changes in the sequence (al,..., a,_l, I). If this number is zer it means that the conditions (8. i9) are satisfied. In the contrary case let i be the 1~ suffix for which ai<o. Then --ai~o, hence --ai--I~>O and A = [0~, ~]P(al)... P(a,_I)C (- i)P(--a,-- ~)P(~)P(a,+I-- I)... P(a,) = ~=[,, --~]P(--al). 9 9 P(--a~_l)P(--a~--I)P(I)P(ai+,-- I)... P(a,). If a~=--I, we can reduce the number of factors P by (8.5). 388 ON THE STRUCTURE OF THE GL~ OF A RING 2 9 If ai+l> I, then there is one less sign change than before and we can apply the induction hypothesis. If a~+t=I , we can combine P(I) with P(ai+~) by (8.5) , unless i~-I =r; in any case the number of sign changes is again diminished by i. There remains the case ai+l<I ; this can only happen when ai+l~<o , hence i+I=r and the same conclusion applies. By induction it follows that the conditions (8. I9) can always be satisfied; moreover only (8.5-7) were used in the process. To prove that the form (8. I8) is unique, let us first assume that A is diagonal. If al>o , then a compa- rison of (I,2)-elements shows that p(al, ..., ar_l)=o , which contradicts Lemma 8. i. If a 1=o, we have P(a~)... P(a,) = DP(o), where D is a diagonal matrix, and comparing (2,2)-elements we find that p(a3, ..., a,_l)=o, which again contradicts Lemma 8. i, unless r~<2. In the latter case we are only left with the possibility a l=a s=o, which was excluded in (8.~o). If al<o, then r= i and A cannot be diagonal. Thus (8. I8) is unique when A is diagonal. In general, let (8. I8) hold and also A=[y,~]P(bl)...P(bs) bl>~o, bi>o (I<j<s). If a,~..-=b~ we can cancel a term and use induction on max(r, s), so we may assume that ar~e bs, say a,.>b~. Writing D1, D2, etc., for diagonal matrices whose exact value is immaterial, we find (8.2I) I : D1P(al)... P(a,.--b~)P(--b,_l)... P(--bl)P(o), hence I = D~P(--al)... P(b,--a,.)P(bs_l)... P(b2)P(b~--I)P(I)P(-- i) = D3P(al)... P(ar--b,-- ~)P(I)P(b~_~-- I)P(b~_2) 9 9 9 P(b2)P(b~-- I)P(I)P(-- I). Since a,.>b~, we have ar--b~--I~>o ; in case this is zero we can combine P(a,_l) with P(I). Then all arguments are strictly positive except possibly the first, last and b1--I. Moreover, a~>o, b~>o. Assume for the moment chat b~>o, then bl--I ~>o and by combining P(b2) with P(i) in case b1--I =o, we obtain a formula in which all arguments except the first and last are >o, and the first is />o. The special case proved shows that this is only possible when there are just two arguments, both zero. Thus we have uniqueness in this case. If b 1 =o, the argument is the same except that the last two factors in (8.eI) are omitted. This completes the proof. The most obvious example of a discretely ordered ring is the ring Z of rational integers. Unfortunately this cannot be extended to the ring of integers in a real algebraic number field, since such a ring is never discretely ordered. If R is any discretely ordered ring, then R[x], the ring of polynomials in an indeterminate over R, is again discretely ordered, if we take as positive polynomials those polynomials which have a positive 389 3 ~ P.M. COHN leading coefficient. E.g. Z [x] is discretely ordered in this way. We shall show below that this ring is not a GE2-ring , but in order to do so we need a refinement of Lemma 8. I. Lemma (8.3). -- Let R be discretely ordered, at, ..., a~eR (r>~ 2), al>~ o, a~> o (1 <i<r) ; then (8.22) p(al, ..., ar)>P(al, ..., a,_l). Proof. -- Write pi=p(al, ..., a~) and assume first that a14: o; then by definition, p, =p,_la, +p~_~, hence Now a,>o, hence ar~> I and pr_x>o, p~_2>o by Lemma 8.1, hence P~>Pr-1, i.e. (8.2e) holds. If al=o , then by (8.I7) , p~=p(a3, ..., a~),p~_l=p(a a, ...,a~_t) and the result still holds. Now let AeGEz(R), where R is any discretely ordered ring. Then by Theorem 8.2, A = [e, ~]P(ql) 9 9 9 P(q,), ql>~o, qi>o (i<i<r). (8.23) Let A=(: ~) and consider c--dqJ" If we compare this expression with (8.23) we see by Lemma 8. i, that b and a--bq, must have the same sign; replacing A by -- A if necessary, we may take this sign to be positive. Then b>a--bq,; now there are three possibilities, according to the sign of qr: Lemma (8.4). -- Let R be a discretely ordered ring, and let AeGE2(R ) be any matrix with first row (a, b), where b > o and assume that A has the form (8.23) with r >1 2. Then (i) /f q~>o, then a>b>o, (ii) /f q,=o, then b>a>o, (iii) /f qr=--c<o, then b>a>--bc. Proof. -- A comparison of (I,2)-elements in (8.23) shows that b =~p(al, ..., at_l) , hence e>o and so ~ may be ignored. If q,>o, we can apply Lemma 8.3 to the first row of A. If q, = o we apply Lemma 8.3 to AP(q~)-t unless r~< 3, when a separate argument is necessary; this may be left to the reader. If q~<o, q~=--c, we have b>a+bc>o and hence the result follows. As an application, let us take any discretely ordered ring R and consider R[x], with the ordering by highest coefficient described earlier. The matrix {I +2x 4 A=~ --x 2 I--2x 390 ON THE STRUCTURE OF THE GL 2 OF A RING 3x is invertible and I +2x>4>o , so if AeGEe, we are in case (i) of Lemma 8. 4 (clearly r>/2 in any representation (8.23)of A). But then q~>o and b>a--bq,>o, i.e. a> bq,> o. Let u be the leading coefficient of q,; then u>o, hence u~> I and so 4u>o, 4u>2; this shows that q, must be of degree zero in x. But for such q~ we have a--bq,>b, which is a contradiction. Hence A cannot be of the form (8.23) and this shows that R [x] cannot be a GEe-ring. In particular, Z[Xl,..., xd] is not a GEe-ring, for any number (>o) of indeterminates. Similarly, free associative rings (i.e. algebras over Z) are not GEe-rings. 9" The commutator quotient structure of GE2(R) and of E~(R). For any group G we shall denote the derived group by G', the commutator quotient group G/G' by G a (i.e. G abelianized) and write x-+x ~ for the natural homomorphism G-+G a. We saw in w 2 that for any ring R, E2(R ) is normal in GE2(R ). When R is universal for GE2, this quotient is abelian; we therefore begin by determining the quotient in this case. Theorem (9. x). -- Let R be any ring universal for GE2. Then (9. x ) GEe(R )/Ee(R ) -~ U(R)t Proof. -- We define a homomorphism f: GE2(R) --> U(R) ~ by the rule E(x) -+ To show that this is well-defined we need only check that the defining relations (2.2-4) are preserved. But this is immediate for (2.2) and (2.3), and for (2.4) it follows from the fact that (c~[~)a=(~,) ". Clearly f maps GE2(R ) onto U(R) a, so to complete the proof we need only show that kerf=Ee(R ). By definition, E(x)ekerf, so the kernel contains Ee(R ). Conversely, if A, given by (2. i i) say, lies in the kernel, then 0r eU(R)'; multiplying by D(~)=E(o)eE(~)E(~-I)E(~), we may suppose that ~ = I, so it only remains to show that [y, I]eE2(R) for all -feU(R)', and this will follow if we prove that [~-a[~-i~{3, I]~E2(R ) for all ~, [3eU(R). Now we saw that D(~)eEe(R ) and (9.2) [~-i~-1~}, I] = D(~-I)D(~-I)D(=})eE2(R ). Hence kerf--Ee(R) and (9.1) follows. Corollary 1. -- In any ring which is universal for GE~, D~(R) aE2(R ) is generated by all matrices D(~), 0cell(R). For if we restrict the homomorphism f: GE2(R ) -+ li(R) ~ constructed in the proof of the theorem to De(R), the kernel is just Da(R) nE2(R ). Now [% ~] maps to i if and only if ,~ eli(R)', but in this case the proof of the theorem shows that [0c, ~] can be written as a product of matrices D(y), yell(R). 391 32 P.M. COHN Cor. I shows that any element AeEy(R ) can be written A----D(~I)D(~2)... D(%)E(al)... E(ar). In particular, any relation in Ey(R ) can be brought to this form, using only (2.2), (2.3) and (9.3) E(x) = which is just the special case of (2.4) where ~ = I. If R is assumed to be quasi-free for GEy, then by the remark just made, we can further reduce this relation to the form (9.4) D(~I) 9 9 9 D(~,) ---- I. Using (9.2) we can finally reduce (9.4) to the form (9-5) [Y1, i]... [y,, I]:I where yieU(R)', yi...y,:i. Thus we obtain Corollary 2. -- In a ring R which is quasi-free for GEy, Ey(R ) is generated by all E(x) (xeR) and if D(e) (eeU(R)) and [y, I] (yEU(R)') are defined by (2.3) and (9.2) respectively, then a complete set of defining relations in terms of these generators is given by E(x) E (o) E (y) = -- E(x+y), E(x)D(c~) = D(~-I)E(~xc~), [Y1, I]... [Yl, I] = I (YieU(R) ', Y1... Y~-~ i). In the case of universal GEy-rings, Theorem 9. i may be thought of as a generaliza- tion (in the case n = 2) of Dieudonnd's determinants over a skew field (cf. [7]). Since the right-hand side of (9. i) is abelian, we obtain Corollary 3. -- For any ring which is universal for GEy, (9.6) Ey(R) GEy(R)'. This corollary can still be improved, as follows: Proposition (9.2). -- For any ring which is universal for GE~ and in which 1 can be written as the sum of two units, (9.7) GE~(R)'-~ E~(R), and hence GEy(R)a~U(R) a. The result follows, once (9.7) is proved, and this is well known (cf. e.g. [2]). It is proved by noting that if a+~---- i, the commutator X-tY-1XY of X-=--[~ -1, I] and Y----E(~-tx)E(o) -t is E(x)E(o)-l; by transposition we obtain E(o)-lE(x) and hence GEy(R )' contains --n(o)n(-- I)n(-- i)E(o)E(o)E(x-- I)= E(o)E(o)D(--,)E(x) = E(x). In particular, when U0(R ) is a field k, we have the Corollary. -- In any k-ring with a degree function, (9.7) holds provided that k has more than two elements. 392 ON THE STRUCTURE OF THE GL z OF A RING It is convenient to consider next the group Ez(R) a. Let R be any discretely normed ring; working mod E2(R )' we have by (2.5) and (2.2), E(x)E(y) = E(x)E(o)E(y)E(o)E(-- I)E(o)E(-- I)E(o)E(-- i) = E(x +Y--3), hence (9.8) E(x)E(y) = E(x +Y--3) (mod E2(R )'). This suggests defining a mapping E2(R)-+R by putting (9.9) E(x)-+x--3. This mapping can be extended to a homomorphism of E~(R) into the additive group of R provided that the defining relations of E2(R ) are preserved. Since E(o)2=--I, we must have --I-+--6 and in particular, (9. IO) I2=o in R. Next the definition of D(0:) shows that (9.xx) D(0:) -+ 2c~ A- ~-1--3, and using this value in (9.3) we find that x-- 3 -4- 2 ~ + 0:- 1__ 3 = 2 0:- 1 + 0:__ 3 + ~x0:-- 3, i.e. ex0:--x=0:--a -1. If we replace x by x+~ -t, this equation reduces to (9. ,2) .x~ = x. Thus in order to be able to define a homomorphism into the additive group of R we must have (9. to) and (9.12). However we can obtain a homomorphism in all cases if we divide out by the appropriate subgroup of R. In order not to complicate the result, let us assume that U(R) is abelian. Then we have Theorem (9.3). -- Let R be a ring which is quasi-free for GE 2 and such that U(R) is abelian, and denote by M the additive subgroup of R generated by 12, all o:xot--x (xeR, c~eU(R)) and all 3(0:-f-i)($-4-i) (0:, ~zU(R)); then (9. I3) E2(R)~ ~ R/M, under the homomorphism defined by (9" I4) E(x) -->x-- 3 (mod M). It should he noted that M is in general not an ideal of R but merely an additive subgroup. Proof. -- It is clear from the remarks preceding the theorem and the definition of M that the first two relations of Theorem 9. I, Cor. 2 are preserved. Moreover, the third relation reduces in the present case to (9-IS) D(0:- 1)D(~- 1)D(~) = I. 5 34 P. M. COHN By definition of M we have ~xo~ - x, hence e-- ~-~ (mod M). Thus D(~) -+2c~ + ~-~--3 --- 3( c~- I) - 3(~ -1- I) (mod M), and the left-hand side of (9- 15) is mapped to 3(o~-- I) -~- 3(~-- I) -~- 3(0~-- I) ~ 3(~ @-~ @-o~--3) - o (mod M). Thus all defining relations of E~(R) are preserved and (9.14) defines indeed a homo- morphism g : Ee(R ) --~R/M. Clearly g is onto, and since the right-hand side is abelian, ker g~E~(R)'; to complete the proof we must establish equality here. By (9.8) we have E(x + 3)E(y + 3) --- E(x +y + 3) (mod E~(R)'), (9.x6) and hence E(xl + 3)..- E(x, + 3) --- E (Y~x~ + 3) (mod E~(R)'). (9.17) If the left-hand side maps to zero, so must the right-hand side, whence ]~xi-= o (mod M). Now we have the identity E(x)E(o)-IE(x)-'F~(o) = E(x)E(o)-~ F~(--x)= -- F,(x) E(--x) ; hence --E(x)E(--x) - I (mod E~(R)'). It follows that D(e ~) = -- D(e)D(-- 0c) = E(~)E(c~-~)E(c~). E(-- ~)E (-- e-~)E(--~) -I (mod P.2 (R)'). Next we have hence E(x)D(e) = D(~-I) E(~x~), E(c~x~)E(x)-l--D(~ ~) --I (mod E2(R )'). (9" 18) If we put x=--3,y=o in (9-I6), we see that E(3) eE2(R)', and so, by (2.8), we obtain E(~x~--x + 3) = E(~x~) E(x)-IE(3) ~E2(R)'. (9" I9 ) (9.18) we also have E(exc~)- E(x), hence E(~) = E(e -1) (mod Ez(R)'), and so By hence E(3~) = E(e)E(o)E(~)E(o)E(~,) -= E(o)~E(~)E(c~-t)E(~) -= D(~), (mod E2(R )'). (9.20) E(3~ ) -D(e) In particular, E(-- 6 + 3) = E(-- 3) = --I, whence (mod E~(R)'). (9.2x) E(I2 +3) =I Further, E(3e~)-D(c~)--D(c~)U(~)-E(3c~)E(3fi) and so E(3 (o~ ~ + o~ + ~+ i) + 3) -= E(3 ~)E(3 ~)E(3 ~)E(I 5) - D(c~)D(c~)D(~) - I, 394 ON THE STRUCTURE OF THE GL= OF A RING by (9.18) and (9.'~I), hence (9.2 o) E(3(~- ~ I)(~ -{- I) 2r-3) -I (mod E2(R)'). Now (9. I9), (9.2 I) and (9.22) show that for any generator x of M, E(x+3) eE=(R)'. By (9. I7) this holds for any xeM. Hence, in particular, if the left-hand side of (9.17) is mapped to zero by g, then the right-hand side lies in E=(R)' and therefore so does the left-hand side. This shows that ker g_cE=(R)' and it completes the proof of Theorem 9.3- The isomorphism (9.13) can be made to play a similar role for E2(R ) as the deter- minant for GL2(R); in this connexion we note that when R is commutative, so that a determinant is defined, then (in case R is discretely normed or discretely ordered) E2(R ) is just the subgroup of GE2(R ) consisting of matrices of determinant I, because every matrix in GE2(R ) is congruent (mod E2(R)) to a matrix of the form [~, 1]. In particular, for a commutative (discretely normed) GE2-ring R we have E=(R) = SL2(R ). As an example, take the ring Z of rational integers. Here M = I2Z and Theorem 9.3 shows that SLe(Z) ~ is cyclic of order i2. Secondly, let I be the ring of Gaussian integers. The only units are 4-i, 4-i and in this case M is generated by 12 and all 2 x and all 3(~ q- I)($ + I). This is just tlhe ideal 2 R, so SL2(I)" is the direct product of two cycles of order two. Suppose that R is a k-ring with a degree-function and moreover, that k is in the centre of R, i.e. that R is a k-algebra. Then M contains x(cr for all xeR, aek (~4=o), and hence coincides with R, unless every non-zero element of k has its square equal to I. In the latter case k cannot have more than 3 elements and then it is easily verified that M = o. Thus we have Corollary 1. -- Let R be a k-algebra with a degree-function, then E~(R)~= o unless k is the field of 2 or 8 elements, in which case E,(R)"=~R. At the other extreme R has no units apart from 4- i ; then it is easily seen that M is the additive subgroup generated by i 2. This is true in particular when R is discretely ordered, and so we have Corollary 2. -- Let R be a discretely ordered ring (or a discretely normed ring in which :k i are the only units), then E2(R)a~R/I 2. For example, if I is the ring of integers in an imaginary quadratic extension of Q, which is not Euclidean, then E2(I) a is the direct product of an infinite cycle and a cycle of order 12. To obtain more precise information about GE~(R) a it is necessary to find out more about E2(R)/GE2(R )' in the cases where Prop. 9.2 does not apply. Let us return to the proof of Theorem 9-3 but drop the assumption that U(R) is abelian. By (2.8), E(~(x--3))E(x--~)-*E(3~) = E(~(x--3)--(x-- 3) + 3~) 395 3 6 P. M. COHN shows that E(3 @ =D(~) (mod GE~(R)'); the proof uses no units apart Now (9" 2o) from ~ and so does not depend on the commutativity of U(R). Since we clearly also have I = E(x)-~[~, I]-XE(x) [~, I] - E(x)-~E(c~x)D(c~) (mod GE2(R ) '), we deduce that E((~-- I)X + 3) ---- I (mod GE~(R)'). (9.~'3) By symmetry we also have E(x(0~--i) ~- 3) -- I (mod GE~(R)'). (9. ~'4) Let N be the ideal of R generated by all c~--i (~eU(R)). We note that N~_M sincee, g., ~x~--x-~(e--i)xe+x(a--i) and I2=--6(--1--1). It is easily seen that the mapping h : E(x)-*x--I (mod N) p~eselves the first two relations of Theorem 9. i, Cor. 2, while the third one becomes trivial (mod GE2(R)'); therefore we have a homomorphism E2(R )/GE2(R )' -+R/N. If the left-hand side of (9. I7) lies in the kernel of this mapping it follows that ~xieN and now (9.23), (9.24) show that the right-hand side of (9-I7) must lie in GE2(R)'. This proves that the kernel is in fact equal to GE2(R )' and we have Theorem (9.4). -- Let R be any ring which is quasi-free for GE 2 and denote by N the ideal generated by all or--I (~eU(R)). Then Ir~ E2(R )/GE~(R) =R/N. E.g., when R=Z, then N=2Z and we obtain the result of Hua and Reiner [ii], that SL2(Z)/GL2(Z )' is cyclic of order two. If I is the ring of Gaussian integers, N=(~ +i)I and SL2(I)/GL~(I )' again is cyclic of order two. Theorems 9. I and 9.4 may be summari'zed as follows: Theorem (9.5). -- Let R be a ring which is quasi-free for GE 2 and denote by N the ideal generated by all ~--I (~EU(R)). Then there is a split exact sequence o -~ R/N -+ GE21'R) a ~ U(R) a -+ o. For the mapping e~-+[~, I] clearly induces a splitting. xo. Generating sets and free products in GE2(R). It is well known that SL2(Z), the two-dimensional unimodular group, can modulo its centre be written as a free product of a 2-cycle and a 3-cycle. It seems to be unusual for GE2(R ) or E~(R) to be expressible as a free product in a non-trivial way, but there are a number of results on large subgroups which have the form of free products (cf. e.g. Nagao [14] ). The question is related to the problem of deciding 396 ON THE STRUCTURE OF THE GL 2 OF A RING when GE2(R) is finitely generated; by means of the normal form for elements in GE2(R ) we shall be able to give an answer to these questions which includes many of the known results as special cases. We begin by discussing the problem of finite generation of GE2(R). Theorem (xo. i ). -- Let R be any ring; if U(R) is finitely generated (as multiplicative group) and R # finitely generated, as U(R)-bimodule, then GE2(R ) #finitely generated. Proof. -- Let U(R) be generated by 71,-.., 7h and R, as U(R)-bimodule, by c1,..., Ck. We already lmow that GE2(R) is generated by all E(a), a~R, and [~, ~], ~, ~EU(R). By (2.2) and (2.4), E(a) can be expressed in terms of E(c,), E(o) and [~, [~], and hence in terms of (io.x) E(o), E(c~) (r= ~, ..., k), [u yj] (i,j: i, ..., h). This shows that GE~(R) is finitely generated. The following is a partial converse: Theorem (xo.~). -- Let R be a ring quasi-free for Gig s and such that GE2(R ) is finitely generated. Then so is U(R) a, and moreover, R is finitely generated, as U(R)-bimodule. Proof. -- Since GE2(R ) is fnitely generated, so is U(R) a, by Theorem 9.5. Moreover, every generating set of GE2(R ) contains a finite subset which is still a gene- rating set. We can therefore find elements cl, ..., ckeR and Y1, -.-, YheU(R) such that the elements (Io. I) generate GI~.z(R ). Thus every element of GE2(R ) can be expressed in the form (xo.2) [~, ~]E(@... E(ar) , where al, ..., ar belong to the U(R)-bimodule generated by Ca, ..., ck. Clearly we may assume that a~4=o (i--~I ..... , r) and using (2.9) we may even take ad~U0(R), provided that we let the a~ range over the U(R)-bimodule generated by i, q, ..., c~. Now take any b~R and consider E(b). If E(b) is expressed in the form (lO.2), we necessarily have r~> i, hence we obtain -- [~, ~]E(al)... E(a~_l)E(a,--b)E(o ) = I. If a,.--br then the argument used to prove Theorem 5.2 shows that r~> 2 and we obtain a contradiction. Therefore ar--bsU0(R ) and it follows that b lies in the U(R)-bimodule generated by I, q, ..., ek; but b was any element of R. This shows that R is finitely generated, as U(R)-bimodule. This theorem shows e.g. that GL2(k[x]), for any field k, is not finitely generated (cf. Nagao [I4] ). It seems likely that Theorem IO. 2 holds for any ring which is uni- versal for GEe. Next we turn to the question of free subgroups and free pioducts. Let R be any ring, and A a subgroup of the additive group of R. Then it is clear that the set B12(A)-={B,2(a) ] aeA} 397 3 8 P. M. COHN is a subgroup of G2(R ) isomorphic to A, and of course the same holds for its conjugate with respect to P(o), which will be denoted by B~a(A ). Theorem (IO.3). -- Let R be a ring which is quasi-free for GE 2 and A, B any two subgroups of the additive group of R which do not meet U(R). Then the subgroup of GE2(R) generated by BI~(A ) and B~I(B ) ~ equal to their.free product. Proof. -- Let F be the subgroup generated by BI~(A ) and Bal(B ). Any element of F has the form f-glhlg2hz...g, hr, where g,~B12(A), h~eB~i(B) and &+I for i0eI, h~+I for i+r. Write g, = B12(a,) = E(ai)E(o) -1, h, = B21(b,) = E(o)-~E(b,), then f has the form (xo.3) zkE(aa)E(ba)... E(a~)E(br), where ai, bi:~o except possibly az or b~; moreover, ai, biCU(R), by hypothesis. In particular, the left-hand side of any relation in F can be brought to the form (io.3). But by hypothesis there are no non-trivial relations of the form (IO. 3) and it follows that F is a free product as asserted. For example, if x, y are elements of R such that nx, t~r ) for all n+ o, then the subgroup generated by Bxe(X ) and B~I(y) is free on these generators. In particular this yields the well known result that for any indeterminate x over a field of characteristic zero, Bx2(X ) and B21(x) generate a free group. Similarly, for any integer m>~ 2, Bxe(m) and B~x(m ) generate a free group. I I. The construction of homomorphisms between general linear groups. Let f: R->S be any homomorphism (a) of rings; clearly this induces a homo- morphism of n� matrix rings f~ : R~-->S~ and since a unit of R~ maps to a unit of S,, we obtain a homomorphism (II.I) f* : GL~(R) -+ GLn(S ). Secondly, let g : R--->S bc an antihomomorphism; this induces in the same way an antihomomorphism of groups g* : GL~(R) --> GL~(S). If we follow this by inversion in GL.(S) we again obtain a homomorphism of groups (I1.2) g: GL~(R) -+ GL,~(S). In particular, when S = R, the automorphisms and antiautomorphisms of P~ give rise in this way to automorphisms of GL~(R). Not all automorphisms arise in this way, however; if a is any homomorphism of GL~(R) into the group of central units of R, then the mapping A-->A.A ~ is an endomorphism of GL,(R), which wilI be called (~) It is understood that such a homomorphism maps the I of R to the i of S. 398 ON THE STRUCTURE OF THE GL 2 OF A RING a central homothety. When we come to study automorphisms of GL,(R) we shall find that for n 1> 3 and a fairly wide class of rings, every automorphism of GL,(R) is obtained by combining/* or ~ with an inner automorphism and a central homothety. For n = 2 the situation is rather different; to describe it we need a Definition. -- Let R, S be any rings. A U-homomorphism f:R-+S is a homo- morphism x~x' of the additive group of R into the additive group of S such that (II,3) I'=I, and (II "4) (,a~)'-=-o~'a'~' for all a~R, o:, ~eU(R). If g is a homomorphism from the additive group of R to that of S which satisfies (I I. 3) and instead of (I 1.4) satisfies (xx.5) (~a~)'=~'a',' for all a~R, v., ~U(R), then g is called a U-antihomomorphism. Clearly a U-homomorphism between fields is just an ordinary homomorphism, and likewise for antihomomorphisms. More generally, we have Proposition (xx.x). -- Let R be a ring which is generated, qua ring, by its units. Then a U-homomorphism of R into an arbitrary ring is a homomorphism and a U-antihomomorphism is an antihomomorphism. Pro@ -- Let x-+x' be: a U-homomorphism and write Ro=:{x~Rl(xa)'=x'a" for all a~R} then by (II,,4), R0 contains U(R) and if x, yeR o then for any aeR, [(x--y) a]'= (xa--ya)'= (xa)'-- (ya)' = x'a'--y'a' = (x'--y') a' = (x--y)'a', hence x--yeR o. Further, (xy)'~-x'y' and for any a~R, (xya) '-= x'(ya)'= x'y' a'= (xy)'a', which shows that xyeR o. Together with (I i. 3) this shows that R 0 is a subring of R and since it contains the units of R, by (II.4), it must be the whole of R. But this means that f:x-+x' is a homomorphism. The proof for antihomomorphisms is similar. Tile form of the defining relations (2.2-4) now yields almost immediately Theorem (ii .2). -- Let R be any ring universal for GI~.2, S any ring and let f: R-+S be any U-homomorphism x-+x'. Then 9 r induces a homomorphism s) /* : Gv. (R) -+ GE (S) by the rule (II.7) E(x)-+E(x'), [~, ~]--+ [~', ~']. 399 4 ~ P. M. COHN If g : R->S is a U-antihomomorphism, then g induces an antihomomorphism g* defined in the same way and hence induces a homomorphi~m (xx. 8) ~ : GE2(R ) --> GE2(S ) by the rule (II "9) E(X) "+ E(x') -1, [9~, ~] ---> [~', ~,]-1. Proof. -- We need only check that the defining relations (2.2-4) and the relations in D~(R) are satisfied and this is clearly the case. Whenf is a homomorphism, the effect off* in (i i. 6) can be described more simply by the rule ,,,,o, (: (:: but it should be noted that for arbitrary U-homomorphisms this is not so. Although there is still a group homomorphism 2'* (given by (II.7)) it does not agree with the mapping (i I. Io), which in general will not be a homomorphism. A similar remark applies to U-antihomomorphisms. Corollary. -- Any U-homomorphism of a universal GE~-ring R defines a homomorphism (i~ .6) of GL2(R ) and any U-antihomomorphism defines a homomorphism (I~ .8) of GL2(R ). When the units in R (together with o) form a field k, a U-homomorphism of R is just a k-bimodule homomorphism such that i -+ i. Thus it induces a monomorphism from k to the image ring, so a U-homomorphism in this case is essentially a k-semilinear mapping. Below we list some special cases of Theorem i i. 2. (i) Let R, S be k-rings and assume that R is a universal GE2-ring with U0(R ) =k. Then any k-semilinear mapping of R into S induces a homomorphism of GL2(R ) into GL2(S). (ii) If R is a k-ring with a weak algorithm, then any k-semilinear mapping to a k-ring S induces a homomorphism of GL~(R). (iii) Any k-linear mapping of a k-ring R with a weak algorithm (into itself) induces an endomorphism of GL2(R ). In the case where R is the ring of polynomials over a commutative field in a single indeterminate, this construction of automorphisms of GL2(R ) is due to Reiner [i5] , [I6]. As an application of the above results, consider the free associative algebra over a commutative field k, on a free generating set which is at most countable. As k-bimodule this is just a vector space of countable dimension over k and so is isomorphic to k[x], the ring of polynomials in a single indeterminate. Hence the GL~ over the free associa- tive algebra is isomorphic to GL2(k[x]). In case :t: I are the only units in R, the statement of Theorem I i . ~2 can be sfightly simplified. Theorem (II "3)" -- Let R be any ring which is universal .for GE2 and in which i are the only units, and let S be ary ring. Then any additive homomorphism Jrom R to S which maps 400 ON THE STRUCTURE OF THE GL 2 OF A RING 4 I I-+I induces a homomorphism of GE2(R ) into GE2(S ) by the rule E(x)---,E(x'), and another one by the rule E(x)-+E(x') -1. Corollary. -- Let R be any ring universal for GEz, bz which 9 I are the only units, then there is an automorphism of GE2(R ) defined by w(x) The statement of the wtrious special cases (in particular the case where R is a universal GE~-ring) may be left to the reader. The result may in particular be applied to the case of discretely ordered rings, where the hypothesis relative to the units in R is automatically satisfied. I2. The analysis of isomorphisms of general linear groups. We now consider the converse problem: Given an isomorphism between GLn(R ) and GLn(S), when is this induced by a mapping f : R-+S? The aim will be to show thatf can be taken to be an (anti)isomorphism or in case n = 2, a U-(anti)isomorphism. We shall limit ourselves to k-rings with a degree-function and to begin with we assume n= 2. .Just as for fields, the case of characteristic 2 has to be treated separately. On the other hand, we do not need to restrict ourselves to GE2-rings, but instead consider rings in which projective modules are free. Lemma (x2. x ). -- Let R be a k-ring with a degree function, where k is afield of characteristic not two, and assume that either of the following conditions is satisfied: (i) R is a GEz-ring, (ii) ever); projective (right) R-module on two generators" is free, then any pair of anticommuting involutions in GL2(R) can be transformed simultaneously by an inner automorphism to the forms [I,--I] and P(o)=(~ ;)respectively. Pro@ -- Let A, B be the given involutions. If (i) holds, then by Theorem 5.5, A is conjugate to [I,--I], because A is non-central. If (ii) holds, consider E= I/2(I+A); clearly E is an idempotent, and since A+ it follows that E4:o, I. If we regard E as acting on R ~, it defines a direct decomposition and by assumption (ii), both the kernel and image of E are free R-modules; taking a suitably adapted basis, we obtain E in diagonal form: E ----- (; o~ and in this coordinate system, A = [I, --, ]. Thus A has been transformed to the form [i,--i], assuming only (i) or (ii). Let B--(: ~), then A-1BA=(__: --~) and byhypothesisthisequals--B, hence a=d-----o. Moreover, B~=I, therefore bc=i, i.e. b, cEU(R) and transforming B by [b, I], which leaves A unchanged, we obtain [c, i]B[b, I]=P(o). Now let R be a k-ring and S a k'-ring, both with a degree-function, where k, k' are 401 42 P. M, COHN fields of characteristic 4= 2 and assume that one of R and S, say S, is either a GE2-ring or has its projective modules free. Let (I2. I) f : GL2(R ) -> GL2(S ) be an isomorphism, and for brevity write Do=[i , --I] (in R or S). Now over any integral domain, --I is characterized as the only central involution, hence (--I)f------I. Since D o and P(o) are a pair of anticommuting involutions in GL2(R), their images underf are again anticommuting involutions in GL2(S), and the same is true of :q(Dof), ~(P(o)J), where ~ is one of the numbers I, --I. Let us make a definite choice of ~, then by composing 3 c with an inner automorphism of GL2(S) we may assume (by Lemma i2. I) that (12.2) D0,f= ~qDo, P(o)f= rjP(o). By (I~.2) the centralizer of D o in GL2(R ) is mapped into the centralizer of D o in GL~(S). But this centralizer is easily seen to be the set of all diagonal matrices, thereforef maps the subgroup D2(R ) onto D~(S). Our next problem is to characterize /T TD_--D-1TD, the triangular matrices T(h)----[~. Writing T=T(h), we clearly \u hi)" have (x2.3) (TD0)2= I, TI)T = TT D for all DeD~(R). (x2.4) I, oHowst theim geU=T s ti . thes meequ tion D----[I, ~], we have UDU=UU D, and equating the (i, I)-terms in this equation we obtain 02.5) b(~--~-l)c= o. If we assume that k' has more than 3 elements, we can find 3~k' such that 324: i, and now (i 2.5) shows that b ---- o or c---- o. In the remaining case k' is the field of 3 elements (because the characteristic is 4: 2) and from (12.3) we obtain (UD0)2= I, whence by equating coefficients, (I2.6) ab=bd, ca=&, aZ--bc=d2--cb= I. Moreover, in this case, T3= I, hence U 3-= I; in detail U = ca + dc d2 -]- cb \ 2 dc 2 d2~ I 2d3 _d_C_2dcb , thus a(ea~--i-]-2bc)=i, d(2d~--i-]-2cb)=i. This shows that a, deU(S) c c_k', i.e. a, d= + i, therefore a 2 = d 2= i and the last equation (I2.6) now shows that bc = o. Again we deduce that either b = o or c = o. 402 ON THE STRUCTURE OF THE GL~ OF A RING Let us assume that b+o, say, then c=o, and by (i2.6) which always holds, we have a2----dZ=I, i.e. (a+I)(a--1)=o. Since S is an integral domain, a----4-1, and similarly d=4-i; moreover, the first equation (12.6) shows that a=d. This then shows that U is of the form and clearly every such matrix satisfies the conditions (12.3) and (12.4) for T. Simi- larlyif b=o but c+o we obtain the form and this matrix again satisfies the conditions (12.3) and (I2.4). Finally, if b=c=o, then all we can conclude is that a=4-I and d=4-I, and thus we have 4-I or 4-D 0. This exhausts all the possibilities. Now f leaves D O fixed except for a scalar factor and it maps diagonal matrices to diagonal matrices, hence/maps any matrix T(h) to a matrix of the form (I 2.7) or (i 2.8). Consider T(I); if T(I)J=T(k) say, where keS, then k+o because f is injective. But for any heR, T(h)f commutes with T(k)=T(i)f, and it must therefore be again of the form (12.7), so in this case the subgroup BI~(R) is mapped into the subgroup 4-B12(S ) byf. The other possibility is that T(I)=B21(k); in this case we modifyJ by composing it with the inner automorphism defined by E(o) = DoP(o ). This leaves (i 2.2) unaffected except to replace ~ by --~, but now T(I)J=E(o)-tB~l(k)E(o)=T(--k), so that this is essentially reduced to the previous case. More precisely, we have shown that by composing f with a suitable inner automorphism we can ensure that (12.2) holds (for ~=4-I) and B12(R ) is mapped into +BI~(S ). As a consequence we can write (x~,. 9) T (x)f= ~(x)T(x~ 4- i}. where x-~.x ~ is a mapping of R into S and x-+r is a mapping of R into Since T(x)T(y)=T(x+y), we have +y)T((x +y)") = ~ = o +ya), hence (x +y)o = x o +yO, (I2. IO) = +y). (X2.XX) Putting y=x in (12.Ii), we find that ~(2x)----~(x)~=1, hence ~(x)=t for all xeR. We saw that x~ for x+o, hence (12. to) shows that the mapping ~ is injective. Repeating the argument with of replaced by f-t (and reading (I2.2) in the opposite 403 44 P. M. COHN direction), we see that G is an isomorphism of the additive groups of R and S. Next we look at E(x): E(x) = DoT(x) P(o ) . Applying f, we obtain E(x)f= r~2DoT(x~ hence E(x)y= E (x~ If we applyf to (2.3) and use (I2. I2) to simplify the result, we get E((a.- t)~ E (0d) E ((0r 1)~ = -- D (0r 1)f. The right-hand side belongs to D2(S), hence so does the left-hand side; but this can only happen if oUeU0(S ). We know that 0r176 and so we find D(~-l)f=__E((~,-1)o__(eo)-~)D(0d)E((a-1)o (eo)-t). Equating the (i, 2)-terms we see that (I2.X3) (~-1)o =(~o)-~, so both sides of this equation may without risk of ambiguity be denoted by ~-~ Using (I2. I3) on the preceding equation we obtain D(~-I).f=D((~~ or replacing ~. by ~-1 and using (I2.I3) again, (12.,4) D(cc)f= D(~). In particular, taking ~= I, we obtain (I2. I5) I ~ I. We can now use Hua's Theorem (cf. e.g. [I], p. 37) and conclude from (I2. IO), (12. I3) and (12. I5) that a is either a homomorphism or an antihomomorphism of k into k'. The same argument applied to f-t shows that a is actually a bijection between and k', and therefore an isomorphism or an anti-isomorphism. If we apply f to (2.4) we obtain D(0~-~176176 0~ ~ -- E(x~ ~ = D (~- ~176 and hence (x2. I6) (~x~) o = ~oxo~ o (x~R, ~U(R)). This relation can actually be used in the proof of Hua's theorem; more generally, we can use it to show, in exactly the same way as in the proof of Hua's theorem, that we have either or (x2. x8) (~x~) ~ = ~~176 ~ for all x~R and ~, ~U(R). Thus e is either a U-isomorphism or a U-anti- isomorphism. Next we consider the effect off on diagonal matrices. Every diagonal matrix can 404 ON THE STRUCTURE OF THE GL~ OF A RING be reduced to the form [i, ~] by multiplying by a matrix D(~), for suitable ~. Now D(~)f is given by (i2. r4) ; we may therefore restrict our attention to [I, 0~]. We know that this is again mapped to a diagonal matrix by f, say (12. I9) [I, ~]f--- [~X, ~]. Since P(o) [I, a]P(o) = [e, ~]-- [e, ~-~] [~, ~], we have [0r ~ 0r -~ [0~ x, o~ ~] = P(o) [o~ x, od'] P (o)= [o~ ~, o~X], and hence (x2. ~,o) ~ = ~~ Using this relation we obtain i.e. [~, Nf=([~, ~-~] [i, ~N)f= b ~ ~-~ [(~)~, (~)o(~)x], (I2.2I) [~, ~]f= [o~o, o~-o(o~)o] (~)x. Suppose now that o is a U-isomorphism; then (I2.2I) states [~, ~]f=[~~ ~](~)~. If we apply this to the equation (i2.22) [~, 0r [I, 0~][i, ~], we obtain [I, 0d }~ (0r x = [I, od]0~z[I, ~3 ~ ~x i.e. (~)~= [i, ~-~ V]~ ~. Equating the (I, 1)-terms in this equation we find (~)~--~, and it can be used to simplify the equation between the (2, 2)-terms to (x2."3) ~o~x=~o. Thus X is a homomorphism of U(R) into U(S), and since ~ maps U(R) onto U(S), (12.23) shows that 0r x centralizes U(S). By (2.4) [~~ 0d] (~)XE((~-ax~) ~) =E(x ~ [~~ t3~ (~I3) x = [~~ ~~176176 ~. This shows again that (0~x~)~176176176 and taking ~-=I, x=yo~ -1 we find o~XE(y ~)=E(y)0r x, which shows that o~ x lies in the centre of S. Thus X is a central homothety, and (I2. i2), (i2.2i) show that f is just the isomorphism induced by the U-isomorphism a, followed by the central homothety X. Finally, a comparison of (i 2.2 I) and (I2.2) shows that ~=(--i) x. Next assume that ~ is a U-anti-isomorphism, then (I 2.2 i) becomes [0~, ~3]f= [0d, 0r176162 x. 405 46 P.M. GOHN If we express ~ in terms of ~r and ~ by means of (12.2o), we can write this relation as [~, p]f= [p-o, ~-o](~p)~. Applying this to (12.22) we find that ~ is a homomorphism of U(R) into the centre of U(S) and as before we can use (2.4) to show that ~ in fact lies iu the centre of S. Thus f is now the isomorphism induced by the U-anti-isomorphism ~, followed by the central homothety ~. The result may be summed up as Theorem (I2.2). -- Let R be a k-ring and S a k'-ring, both with a degree-function, where k and k' are any fields of characteristic 4:2 and S is either a GEz-ring or all projective S-modules on two generators are free. Then every isomorphism between GL2(R ) and GL2(S ) is obtained by taking the isomorphism induced by a U-isomorphism or U-anti-isomorphism, followed by a central homothety and an inner automorphism. This includes the result of Reiner [16] when S=R=k[x] and earlier results of Schreier-v.d. Waerden and Hua (cf. [8] and the references given there). IfR is any k-ring, then the characteristic ofk is =t= 2 if and only if GL~(R) contains a central involution. Hence if we are given an isomorphism f : GL2(R)~GL2(S), where R is a k-ring and S a k'-ring, then k, k' both have characteristic =~ 2 or both = 2. In order to deal with the latter case we nced a Lemma analogous to Lemma 12. i. Whereas Lemma 12. I shows that in characteristic not two (and under the given hypo- theses) the subgroups of GL2(R ) of the type of the dihedral group of order eight form a single conjugacy-class, the next lemma establishes a corresponding fact for subgroups of the type of the symmetric group of degree 3, in the case of characteristic 2. Lemma (x2.3). -- Let R be a k-ring with a degree-function, where k is a field of charac- teristic 2, and assume further that R is a GEz-ring. Then any pair of involutions in GL2(R ) whose product has order 3, can be transformed simultaneously by an inner automorphism to the forms T(I)(=BI~(I)) , P(o) respectively. Proof. -- Let A, BeGL2(R ) be the given involutions and write C=AB. By Prop. 5.4 and the remark following it, C can be brought to one of the forms (x2.24) [I, ~]E(a), [0~, ~]E(o)E(b). Suppose that C has the second of these forms; comparing terms in the equation C 3 = I, we see that ~3 = ~.~ = I, b0~ 2 + ~b~ + ~Zb = o. If we transform by E(~b~) -1, we therefore obtain E(~be) [,, ~]E(o)E(b)E(o)E(~'b00E(o)= [~, e]E(~be2)E(o)E(b)E(o)E(~ebo~)E(o) -- [~, ~]E(b+ ~Zb,+ ~b,2)E(o) Thus C has been transformed to diagonal form. If C has the first form in (i2.24) , a 3 thcequation C3=I showsthat [~=a 2, I. When a=~ I, it follows that a2+a+ I =o, 406 ON THE STRUCTURE OF THE GL,~ OF A RING 47 transforn ng by (i i) I ~ we reduce C to the form [~, I]. Thus C has been reduced to either of the following forms We assert that in fact C can always be reduced to the first of these forms. Since C has order 3, we have ,a=~a__ I and ,, ~ are not both i in the second form (i2.25). Supposethat-=I, sayandlet A=( a bd), thenbyequating(I,I)-termsintheequa- tions A2=(AC)2=I, we find aS+bc=a2+b~c=i, hence b(~--I)C=O; therefore b or c must vanish, say b=o. Then a----d--i, and equating (2,2)-terms in (AC)8=I we find ~2= I, i.e. ~---= I, which is a contradiction; the same reasoning applies if c=o. Thus neither :c nor ~ in (I2.25) can be I and they therefore satisfy the equation (x2.26) xS + x + I =o. Assume first that this equation has a root in the centre of k, say (0. Then (.--(0)(.--(08 ) =% hence .----(0 or .=(08, and likewise for ~. If ~=., then C is a scalar matrix; leaving this case aside for the moment, so as not to interrupt the argument, we may assume that ~4=~; it f~176 that ~--"-t, andtransf~ ~) we obtain the first form in (12.25) for C. If (I2.'26) has no roots in the centre Z of k, it must be irreducible over Z and so all its roots in k are conjugate (Herstein [io]). Hence there exists yek such that y-l~y = e-1 and transforming [,, ~] by [i, y] we are reduced to the previous case. Thus C has n~ been reduced t~ the f~ ( i I ;) (with the exception noted). Now by hypothesis, C=AB, C-I=BA, therefore (x2.27) CB = A----- BC -1 . Taking B=(; ~) and using the expression found for C we obtain by equating components in (12.27) and simplifying, z=u, u+v+w=o. IfweequatetermsinB2=I, wefind thatuv=vu, u2+vw=I. Moreover, A=(: ;); and by Theorem 5.5, A is conjugate to T(h), for some heR. Thus there exists an inver- tible matrix P=(c a b) such that (I ~ . 28) AP = PT(h). Equating coefficients, we obtain the equations va + wc = a, vb + wd = ah + b, ua + vc = c, ub + vd = bh + d, 407 48 P.M. COHN or after some simplification, ( v -1- i ) b + wd = ah, (I2.29) i (V'~- I)a=ZOC, ub+(v+ I)d=bh. Now (12.28) may be written as (A+I/P=P( ~ ~). The first row of this matrix equation reads (I2.30) (V-~- I, w)P=(o, ah), hence (I2.31) (/) -~- I, W)=(O, ah)P -t. From the first two equations (i2.29) , we have (x2.32) (v+ I)C +w(a+c)=a, and by (i 2.3 I), ah is a common left factor of v + I and w, therefore a = ahk, whick shows h to be a unit, h=-h say. Combining (12.32) and the third equation of (12.29) we obtain an equation which in matrix form can be written Using (i2.3o), we may write this as ,,,-1[ b+c~ (o, If a=o, then c=t=o and the first two equations (12.29) show that w=o, v==i, anc hence u= i. In this case transformation by C -1 achieves the desired reduction If a~= o, then a~ + o, and from the last equation written it follows that p-l{ b+c~q for keR. \ d + (a + c)~qJ = t o ] some Hence d+(a+c)~]= \o]' Inserting these values in P, we find that thus b=c~q+ak, d=(a+c)'~+ck. ak + c~q ~ c I k p [ a c ~ to be invertible, evidently it commutes witt This shows the matrix 1=/c\ a+c C=(: ;) and transforming a by PI we obtain T(x ). The same transformatior reduces B=AC to the form P(o), as we wished to show. 408 ON THE STRUCTURE OF THE GL z OF A RING It only remains to show that the case C =coI, where co is a root of (I2.26) in the centre of k, cannot occur. Let us assume the contrary and write R- ={x Rlxco =co x}. For brevity the elements in R + will be called symmetric and the elements in R- skew. By hypothesis, every element of k is symmetric; further, every element of R can be expressed as the sum of a symmetric element and a skew element in just one way, for if (I=.33) x=x +@x- (x~t, z=~), then (I2.34) cox=x + co q- x-co ~, and solving the equations (12.33-4) we find x + ----- xco 2 -~ cox, x- ----- xco -[- cox; thus only one choice is possible for x +, x- and this choice clearly satisfies (I2.33). By (i2.27) we have coB = Bco 2, hence all the elements of B are skew. Now B is invertible and hence of the form [~, ~]E(ql)... E(q,). Denoting the first row of B by' (a, b), we have a = bq~ + a', (I2.35) where d(a')<d(b), by Lemma 5-i. If we equate the symmetric components here we obtain b(qr)- +(a') + =o. But d((a ' ) + )<~d(a ' )<d(b), hence (a') + =(qr)--=O and so a' is skew and q~ is symmetric. Therefore BE(qr) -1 consists entirely of skew elements; by induction on r we conclude that [~, ~] consists of skew elements, but this contradicts the fact that k_c R +. This completes the proof of Lemma ~2.3. With the help of this lemma it is now an easy matter to obtain an analogue of Theorem I2.2 for the case of characteristic 2. Let f: GL2(R) ~ GL2(S ) be an isomorphism, where R is a k-ring and S is a k'-ring, both with a degree-function, k and k' are fields of characteristic 2 and moreover S is a GE~-ring. In GL2(R ) the matrices T(I), P(o) form a pair of involutions whose product has order 3, hence so do their images in GL2(S), and by Lemma I2.3 we may therefore assume (by combiningf with a suitable inner automorphism of GL2(S)), (i2.36) T(I)f= T(I), P(o)f= P(o). 409 50 P.M. GOHN The subgroup B12(R ) of GL~(R) may be characterized as the maximal abelian subgroup of exponent 2 in the centralizer ofT(i) in GL2(R ) (1). It is therefore mapped to B12(S), the maximal abelian subgroup of exponent 2 in the centralizer of T(I) in GL2(S ). Thus there is a mapping a : R-~S such that T(x)f= T(x ~ xsR. By (2.2), we have (x +y) ~ = x ~ +y~ (I2.37) and by (I2.36), i~ I. (x2.38) Now a diagonal matrix D in GL2(R ) is characterized by the fact that both D and P(o)DP(o) normalize B12(R). This shows that f maps diagonal matrices over R to diagonal matrices over S; we can now follow the proof of Theorem i2.2 exactly and finally obtain Theorem (x2.4). -- Let R be a k-ring and S a k'-ring, both with a degree-function, where k, k' are any fields of characteristic 2 and S is a GE2-ring. Then every isomorphism between GL2(R ) and GL2(S ) is obtained by taking the isomorphism induced by a U-isomorphism or a U-anti- isomorphism, followed by a central homothety and an inner automorphism. In conclusion we briefly discuss the isomorphisms of GL,(R). It turns out that by a method similar to that used in proving Theorem I2.2, we obtain Theorem (i2.5). -- Let R be a k-ring and S a F-ring, both with a degree-function, where k and k' are fields of characteristic ~ 2, and assume further that everr finitely generated projective S-module is free. Then every isomorphism between GL,(R) and GL,(S) (for n >~ 3) is obtained by taking an isomorphism or anti-isomorphism.from R to S, followed by a central homothety and an hmer automorphism. Proof. -- GL,(R) contains a system J of 2" commuting involutions, namely [~: I, ..., :t: i]; moreover, the symmetric group Z of degree n acts on this set by permu- tations: there are n+I orbits, of (;)elements respectively (k=o, i, ..., n). The isomorphism f transforms J into a set of 2" commuting involutions in GL,(S). Now if P is any involution in GLn(S), then E -. 1/2(1-1-P) is an idempotent, so that we get a set of 2 ~ commuting idempotents in the matrix ring S,. Since projective S-modules are free, any idempotent can be diagonalized, and likewise any commuting set of idem- potents can be transformed simultaneously to diagonal form, with diagonal elements :5 I. By applying this transformation we thus bring the 2 n commuting idempotents to the form E=[el,...,e,], where e i=o or i. Now P=2E--I is again an involution, so that we have a system of 2" commuting involutions in diagonal form in GL,(S). More precisely, by applying a suitable inner automorphism, we may assume that for any (a) This subgroup may also be described as the set of all involutions in the centralizer of T(I), together with I. 410 ON THE STRUCTURE OF THE GL 2 OF A RING 5x involution P=[~I, "" ", gn], *i=-t-I, 1)j iS again diagonal, with on the diagonal. Ifk of the ~i are +i and the rest are --i, P is said to be of type (k,n--k) ora (k, n--k)-involution. The subgroup of X centralizing a (k, n -- k) -involution has the form Xk� hence ifP is of type (k, n--k), then P/is of type (k, n--k) or (n--k, k), i.e. either P/ or --Pf is of type (k,n--k). Consider the n (i, n--I)-involutions P1, ..-, P,; they form an orbit under Z, hence P1/, --., P,J are conjugate and so they are all of type (1, n--i) or (n--l, I). By combining/with a suitable inner automorphism we may thus assume that (I2.39) PJ=~PI (~ =+:, i= I, ..., n). Since every involution in fir is a product of P~'s, we have, for any involution PGJ, P/=~pP where ~e=~q k if P has type (k, n--k). Let So be the permutation matrix corresponding to the permutation ~eX; then S~-IPiSo = Pio, hence applying / and noting (I2.39), we obtain (So/)-*P, (So/) = P,o. Therefore So(So/) -* commutes with Pi for i= i, ..., n and hence is diagonal, say (12.4o) So/=DoSo. Consider the n-cycle --(1 2 ... n) and let Dp=[%, ...,%]; then theequa- don S~=I shows that (I2.4I) O~nO~n_ t . . . (Xl= I. If we transform the images under / by a fixed diagonal matrix T = ['(1, 9 9 Y,], the PGJ remain unchanged and (12.4 o) takes on the form Sa/= D'o So, where D'~ = T-1Do. S~TS2 a. In particular, we have D'p = [u 7~-1%71, ..., 7~-1%`(,_1]. Our aim will be to choose ~==kI and Yi(i=I, ...,n--I) such that (I2.42) y~-1~iy~_l= ~ (i-= I, ..., n; go-----Ya= I). If this is to hold, then by (12.41) we must have yl=~l~, ~'2=0~2y1~---r '1, ..,, 7n_l=~n_l... IXl~ n-t, yn=~ n= I; conversely, these equations will ensure that (12.42) holds. Now these equations can always be solved if n is even, while for odd n they can be solved provided that ~ = I. To sum up, we can always transform by an inner automorphism (induced by a diagonal matrix) such that for p=(r ~ ... n), (:2.4 ~ ) reduces to ( 2.4a) so/=z(p)so, where X is a linear character of Y,. 411 52 P.M. COHN Next consider a transposition, say z=(I2); if Dr=[~l, ..., [5,], the relation 2 2 St-- I shows that ~1~2-- I, ~i = I (i/> 3). If we put Sr into diagonal form for a moment we see that it is a (I, n--i)-involution, and hence Srfis of type (i, n--i) or (n--i, i); this means ~3 = ~4 .... -- ~5, (-- + i). Thus D,=[~, ~-1, 8, ..., 8] (~=+i). Now the equation (i 2 ... n)=(n n--i)(n--i n--2) ... (3 2)(3 I) shows that s0 = "-3... st, where S p denotes the transform of S by Sp. Applying J and equating diagonal terms, we find that If n is odd, Z(P) = I and these equations reduce to In this case D r = 3I is a scalar matrix and since ~, -r generate Z, every D o is scalar, in fact, 02.45) soy= Z(~)So, where Z(~) is a linear character of Z, the identity or the alternating character according as 8=I or =--i. Ifnis even, the equations (12.44) reduce to (= and in this case ~ is still at our disposal. We now choose ~--= Z(P); then D r is again a scalar matrix and so (12.45) holds in this case too. Now the diagonal matrices may be characterized as the set of matrices centralized by the P's; consider the set centralized by 02.46) ..., Clearly this is the set of all A~-[d3, ..., d,], where AeGL2(R). Hence the set centralized by the elements (I2.4 6) and the permutations not involving I or 2 is A@XI,_ 2 (Zek). ThusJ'induces a mapping A~-I,_~ ~ Af~-X(A)I,_e, wherefis an isomorphism GL2(R)-§ GL2(S). Transforming both sides by S o (for ~eX) we see that a corresponding formula holds for other rows and columns. Now by Theorem 12.2, fis of the form fl Jeff, where fl is induced by a U-isomorphism or a U-anti-isomorphism ?, say the former, f2 is a central homothety and f3 is an inner automorphism. Thus f~ maps Bij(x ) to B0.(x~), and since (B~z(x), B23(Y)) = B13(xy ) (where (A, B)=A-~B-~AB), we have (B~2(x~), Bea(y~)) = Bla((xy)~), i.e. 412 ON THE STRUCTURE OF THE GL 2 OF A RING Thus ? is in fact an isomorphism; similarly a U-anti-isomorphism is shows to be an anti- isomorphism. By taking commutators with elements of the form I, ~-B we see that k(A) lies in the centre of S; dividing by ;~(A) we thus have a homomorphism Aq- In_~ ~ 0(A)A~ q- In_2, and by definition, 0(A) = I on involutions. Further, [0r I, I, ..., I] ---> [0(0r ~, 0(0r I, ..., I]; permuting the second and third rows and columns (which does not change the left- hand side) we see that 0([~., I])= I. Therefore 0 = I and the result follows. REFERENCES [i] E. ARTIN, Ge0m~tr/c Algebra (New York, x957). [2] H. BAss, K-theory and stable algebra, Publ. math. LH.E.S., n ~ 22, Paris (1964). [3] H. BAss, Projective modules over free groups are free, 07. of Algebra, I (I964) , 367-373 . [4] P- M. COH~, On a generalization of the Euclidean algorithm, Proc. Cambridge Phil. Soc., 57 (I96Q, 18"3o- [5] P. M. COHt% Rings with a weak algorithm, Trans. Am~r. Math. Soc., Io9 (I963), 332-356. [6] P. M. COHN, Free ideal rings, J. of Algebra, i (1964), 47-69. [7] J. DIEUDONN~, Les ddterminants sur un corps non commutatif, Bull. Soc. Math. France, 71 (I943) , 27-45. [8] J. DmUDONN~, La gdomdtrie des groupes classiques, Ergeb. d. Math., n o 5 (Berlin, I955). [9] G. H. HARDY et E. M. WRIGrIT, Introduction to the theory of numbers, 2nd ed. (Oxford, I945). [io] I. N. HERSTEm, An elementary proof of a theorem of Jaeobson, Duke Math. 07., 2I (I954), 45-48. [ I I ] L. K. HUA et I. REINER, Automorphisms of the unimodular group, Trans. Amer. Math. Soe., 71 ( 1951 ), 331-348. [I2] W. KLINOENBERG, Die Struktur der linearen Gruppen fiber einem nichtkommutativen lokalen Ring, Archly d. Math., 13 (I962), 73-8L [i3] J. LANDIN et I. R~Im~R, Automorphisms of the two-dimensional general linear group over a Euclidean ring, Proc. Amer. Math. Soc., 9 (I958), 2~ [i4] H. NAOAO, On GL(~, K[x]), J. Inst. Polytech. Osaka City Univ., Ser. A, Io (I959), II7-I2I. [I5] L REINER, A theorem on continued fractions, Proc. Amer. Math. Soc., 8 (I957), IIII-III 3. [I6] I. P, mINER, A new type of automorphism of the general linear group over a ring, Ann. of Math., 66 (i957) , 461-466. [I7] B. L. v. d. WAERDEN, Moderne Algebra, I (Berlin, i937). [18] J. M. H. WEDDERBURN, Non-commutative domains of integrity, 07. reine u. angew. Math., I67 (1932), 129-I4I. Queen Mary College, University of London. Manuscrit refu le 15 juillet 1965. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/on-the-structure-of-thegl-2-of-a-ring-K2DTEQ0E4k

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References (15)

L. Hua, I. Reiner (1951)
Automorphisms of the unimodular group
Transactions of the American Mathematical Society, 71
H. Bass (1964)
Projective modules over free groups are free
Journal of Algebra, 1
I. Reiner (1957)
A New Type of Automorphism of the General Linear Group Over a Ring
Annals of Mathematics, 66
I. Herstein (1954)
An elementary proof of a theorem of Jacobson
Duke Mathematical Journal, 21
J. Wedderburn (1932)
Non-commutative domains of integrity.
Journal für die reine und angewandte Mathematik (Crelles Journal), 1932
P. Cohn (1963)
Rings with a weak algorithm
Transactions of the American Mathematical Society, 109
H. Bass (1964)
K-Theory and stable algebra
Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 22
P. Cohn (1964)
Free ideal rings
Journal of Algebra, 1
W. Klingenberg (1962)
Die Struktur der linearen Gruppe über einem nichtkommutativen lokalen Ring
Archiv der Mathematik, 13
P. Cohn (1961)
On a Generalization of the Euclidean Algorithm
Mathematical Proceedings of the Cambridge Philosophical Society, 57
Joseph Landin, I. Reiner (1958)
Automorphisms of the two-dimensional general linear group over a Euclidean ring
, 9
H. Nagao (1959)
On GL(2, K[x])
, 10
P. M. Cohn (1961)
On a generalization of the Euclidean algorithm
Proc. Cambridge Phil. Soc., 57
I. Reiner (1957)
A theorem on continued fractions
, 8
J. Dieudonne (1943)
Les déterminants sur un corps non commutatif
Bulletin de la Société Mathématique de France, 71

Publisher: Springer Journals
Copyright: Copyright © 1966 by Publications mathématiques de l’I.H.É.S
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/BF02684355
Publisher site: See Article on Publisher Site

Abstract

by P. M. COHN CONTENTS ]3Attt~S I, INTRODUCTION ............................................................................ 5 2. Elementary matrices over an arbitrary ring ................................................. 7 3, Direct and free. products of GlE-rings ....................................................... 1o 4" GL~(R) for a local ring ................................................................... [4 5- Discretely normed rings ................................................................... t 5 6. Rings of algebraic integers and algebraic functions ........................................... 2 i 7- k-rlngs with a degree-function .............................................................. ~4 8. Discretely ordered rings ................................................................... ",6 9. The commutator quotient structure of GE:~(R) and E~(R) .................................... 3I Io. Generating sets and free products in GF~(R) ................................................ 36 x z. The construction of homomorphisms between general linear groups ............................ 38 1,,. The analysis of isomorphlsms of general linear groups ........................................ 4t REr~:a~z.'ccss .................................................................................. 53 x. Introduction. The general linear group over a field, its subgroups and automorphisms, have been studied fairly extensively, even when the field is skew (cf. [8] and the ret~rences given there), but little is known about the general linear groups over an arbiuary ring. Now in the case of fields, the starting point is the observation that every" invertible matrix is a product of elementary matrices; this suggests that in studying GL,(R) it is best to confine attention at first to rings which share this property, and we therefore define a generalized Euclidean ring, or GE-ring for short, as an integral domain (not necessarily commutative) such that for all n, GE, : Every invertible n × :~ matrix is a product q] elementary n × n matrices. Examples of GE-rings are (i) the classical Euclidean rings (cf. [17]), (ii) rings with a weak algorithm [5], in particular ti'ee associative algebra~ over a commutative field, and (iii) free products of GE-rings which are also semifirs (cf [6] and § 3 below); this includes in particular the group algebras of free groups and free products of skew fields. 365 6 P. M. COHN The present paper is a study of GL~(R), for various types of rings, and in particular GE-rings; as in the case of fields, there is a basic difference between the case of dimension 2 and dimension gleater than 2, so that it is reasonable to begin by concentrating attention on the former. For most of our results we shall find it enough to assume GE~ for n = 2 only; in fact we can even dispense with GE 2 by limiting ourselves to the subgroup of GL~(R) generated by the elementary matrices, and the results will usually be stated in this form. This more general point of view is actually forced on us when we come to give examples of rings that are not GE-rings. Compared with Euclidean rings, GE-rings have a more intrinsic definition, in that no norm function is involved. But for a closer study of GL~(R) we shall find some sort of norm function on R of great use. Most of our results will apply to discretely normed rings, defined in w 5 and to discretely ordered rings defined in w 8. It is the presence of the norm (or the ordering) which usually enables us to decide whether a given ring satisfies GE,. As an application we shall show that the ring of polynomials in any number of indeterminates with integer coefficients, and the ring of polynomials in at least two indeterminates with coefficients in a field, are not GE-rings (w 5). Further it is shown that the ring of algebraic integers in an imaginary quadratic number field is a GE-ring if and only if the field is Euclidean with respect to the usual norm (w 6). In particular (1) taking the integers in O (~--~) we obtain a principal ideal domain which is not a GE-ring (clearly this says rather more than the usual assertion that this ring is not Euclidean). The main tool of the paper is an explicit presentation of GL~(R) for suitable GE-rings, called universal GE-rings (w 3). As examples of such rings we have fields or more generally local rings (w 4); moreover, any GE-ring which is either discretely normed or discretely ordered is a universal GE-ring. This presentation brings out in a particularly clear form the extent to which GL~(R) is independent of the multiplicative structure of R, and which accounts for the crucial difference between GL~(R) for n = 2 and n>2. Thus a generalization of a ring homomorphism, the U-homomorphism, is introduced in w i I and it is shown that any U-homomorphism between two universal GE-rings induces a homomorphism between their GL~-groups. As a rather striking illustration one has the result that for any free associative algebra A over a commutative field F on at most countably many free generators, GL~(A)-~GL2(F[x]). The auto- morphisms of GL2(F[x]) constructed by Reiner in [16] can also be obtained very simply from this point of view. In the other direction the isomorphisms between the GL~-groups of GE-rings are studied under the assumption that the rings have a degree function defined on them such that all the elements of degree zero are units. It is shown that any such isomorphism can be built up by taking a U-isomorphism (or a U-anti-isomorphism), following it by a cen- (1) This answers the question raised by various authors (I. Reiner, J.-P. Serre) whether every Dedekind ring, or more particularly, every principal ideal domain, is a GE-ring. 366 ON THE STRUCTURE OF THE GL~ OF A RING tral homothety and an inner automorphism (just as in the case of fields, cf. [8], where the notion of U-(anti-)isomorphism reduces to the ordinary (anti-)isomorphism). The proof is based on the fact that in a ring of the type named, all dihedral subgroups of order eight are conjugate, in case the characteristic of the underlying field is ~ 2, while in charac- teristic 2 all subgroups of the type of the symmetric group of degree 3 are conjugate. This last fact rests on a surprisingly delicate argument (Lemma 12.3) and it would be of interest to have a simpler proof. The presentation introduced in w 2 can also be used to study the commutator quotient structure of GL2(R ). This is done in w 9 where an analogue of SLy(R) is defined, denoted by E2(R ) and it is shown that for a universal GE-ring, GL,(R)/E2(R)~--~U(R) a, where U(R) is the group of units of R and for any group G, G ~ = G/G' is G made abelian. Further, if R is a discretely normed ring in which U(R) is commutative, then l~.2(R)"-~ R/M, where M is a certain additive subgroup of R deter- mined by the units of R (for a precise statement see Theorem 9.3). A number of other applications and generalizations of known results are given in w Io. 2. Elementary matrices over an arbitrary ring. For a Euclidean ring R it is well known that GL2(R), the group of invertible 2 � 2 matrices over R, is generated by the elementary matrices (; ;) (; :) (o ;) where ~, ~, aeR and ~, [~ are units. Our object is to study, for any ring R, the group generated by these matrices; we shall denote this group by GE2(R ). In particular, we shall be interested in rings R for which GE2(R ) =GL2(R); such rings will be called GE2-rings. Clearly they include the GE-rings mentioned in w I. Let R be any ring (always associative, with I) ; generally we shall denote arbitrary elements of R by latin letters and reserve greek letters for invertible elements of R. The group of units of R is denoted by U(R) and Uo(R)-~U(R)u{o }. Further we set and denote by D=D2(R ) the group of 2� invertible diagonal matrices over R and by E=E2(R) the group generated by all E(a), aeR. Thus GE~(R) is the group generated by D and E. If we write Bq (a)= I-t-aeq, where e# are the usual matrix units, then Bx2(a)=E(--a)E(o) -1, S,l(a) =- E(o)-lE(a), and E(o) ~--~ B12(I) B21 (- I)BI~(I), E(a) = B12(-- a)E(o) ; these equations show that E2(R) is just the group generated by all B~(a) (aeR, i4:j). However, we shall mainly keep to the generators E(a), since the defining relations are 367 8 P. M. COHN expressed rather more easily in terms c t them. We have the following relations between the matrices (2. I) : (2.2) E(~)E(o)E(y) =--E(x +y), (2.3) E(=)E(~-I)E(~) = --D(e) x, yeR, (2.4) E(x) [cr [B] = [[3, ~]E(~-'x=), 0r ~eU(R). Their verification is elementary and is left to the reader. These three relations are basic for much of what follows, for as we shall see, there are large classes of rings in which the relations (2.2)-(2.4), together with the relations in D~(R), form a complete set ot defining relations for GE~(R). A ring which has this property is said to be universai for GE2; of particular interest are the GE2-rings which are universal for GE2, or the universal GE2-rings, as we shall call them. Thus a universal GE~-ring is characterized by the property that GL2(R ) is generated by the matrices (2. I), with (2.2-4) and the relations of D 2 as a complete set of defining relations. The first task is to derive a number of consequences of (2.2-4) : Putting x =y = 0 in (2.2) and ~=4-I in (2.3), we find (2.5) E(o)Z=--I, E(I) 3=-I, E(--I) 3=I- Putting y=--x in (2.2) and using (2.5) we get E(x) -1 ~- E(o)E(--x)E(o) ; (2.6) explicitly we have ~ -3' By (2.6) we have (2.7) E(x) E(y)-l= E(x--y) E(o)-l=--E(x--y)E(o). From (2.7) we easily obtain the following generalization of (2.2): (2. s) E (x) E (y)- 1E (z) = E (x--y + z). If ~ is any unit in R and x, yeR, then by (2.3), E(x) E (~- I) E(y) _-- -- E(x) E (=)- 1D(=) E (~)- ~ E(y) = -- E (x-- =) E(o) D(~) E(o) E(y-- =) --__ E(x--=)D(~-a)E(y--=), where we have used (e.3), (2.7), (2.4) and (e.5). Replacing ~ by ~-1, we find (2.9) E(x) E(~)E(y) =E(x--~-~)D(=)E(y-- ~-I). Finally if~, ~ are any units, and x,y~R, then E(x)E(~ -~ I)E(~ q- I)E(y) = E(x)E(e)E(o)E(I)2E(o)E(~)E(y) ---- -- E(x)E(~)E(-- I)E(~)E(y) ----- -- E(x-- ~-I)D(~)E( - I -- ~-I-- ~-I)D(~)E(y--~-'), 368 ON THE STRUCTURE OF THE GL~ OF A RING where we have used (2.6), (2.5) and (2.9) in turn. Hence (~z.xo) E(x)E(~ + I)E({3.-t- i)E(y) = -- E(x-- ~-~)D(e)E( - ~--~-l--~-~)D(~)E(y--~-l). We conclude this section by showing that in any ring, De normalizes Ig e and by obtaining a certain standard form for the elements of GE e which will later be shown to be unique, for suitably restricted rings. Proposition (2. x). -- In any ring R, GE2(R ) = DE = ED and E is normal in GE~(R). Proof. --By (2.4), (2.5) and (2.3), [o~, ~]-'E(x)[~, ~] = [,-~, ~-~]E(~-~x~) = D(,-~)E(~-~x~) Thus IgDclg and since GEe(R ) is generated by D and E, we have E<] GEe(R ) and the result follows. We note that in general DnE~= i, so that we do not have a semi-direct product. In fact, by (2.5) and (2.3), D(o~)eDnE for all ,EU(R); later (in w we shall find conditions under which the subgroup generated by the D(e) is exactly DnE. By definition, every element of GEe(R ) is a product of matrices [0~, ~], E(x) and E(x) -1. Now by (2.6) any factor E(x) -1 can be replaced by a product of E's, so it follows from Proposition 2. I that every element A of GEe(R ) has the form (2. x,) A= [~, ~]E(a~)... E(a,). If for some i in i<i<r, ai=o, then we can shorten (2. II) by using (2.2); if aieU(R ) we can use (2.9) and (2.4) to shorten (2. i I). Thus after a finite number of steps we reach a form (2.ii) which cannot be shortened in this way and then a~r ) for i <i<r. Moreover, when r =-- 2, we may assume by (2.5) that al, a2 do not both vanish. Such an expression for A is s:dd to be a standard form. We note that only (2.2-4) and their consequences have been used in obtaining the standard form. Our results may now be summed up as follows: Theorem (2.2). -- Let R be any ring and denote by GEe(R ) the group generated by all matrices (., ~U(R)) and E(a)=(__: Io) (aeR), and set D(o~) = [o~. o~-1]. These generators satisfy the relations: E(x) E(o) E(y) ---- --E(x +y), (2.2) (2.a) E(,z) E(.-~)E(~) = -- D(.), E(x) = (2.4) "2 Io P.M. GOHN These relations, together with those in the group D(R) generated by the [% [~] imply E(o)'=--l, g(~)~=--l, E(--I)3=I, (2.5) (2.6) E(x) -t = E(o)E(--x)E(o), E(x) E (y) -t ---- E (x--y) E (o) -~ -- -- E (x--y) E (o), (2.7) (2.8) E(x)E(y)-tE(z) -= E(x--y + z), E(x)E(~-a)E(y) = E(x--cz)D(a-t)E(y-- e), (~'.9) (2.xo) E(x)E(~+ I)E(~ + i)E(y) = -- E(x--~-X)D(~)E( - i --e-~--~-t)D(~)E(~--~-1). Moreover, they imply that the group E2(R) generated by all E(a) /s normal in GE2(R ) and that eve O, element A of GE2(R ) can be expressed in standard form (2.II) A = [=, NE(al). 9 9 E(a,), where ~, ~zU(R), a~zR and such that a~CU0(R ) for I<i<r and al, a 2 are not both zero in case r = 2. In some rings it is possible to shorten (2. i I) still further by an application of (2. i 0), but for many rings such a reduction is impossible and (2. r I ) actually represents a normal form for the dements of GE~(R) (cf. w167 4 and 7). Such a ring is said to have a unique standard form for GE 2. We shall also meet rings R for which the only relation in GE2(R ) of the form W = I, where W is a word in standard form, is the trivial rela- tion I = I. Such a ring is said to be quasi-free for GE 2. It is clear that of the properties: (i) R has a unique standard form for GE2, (ii) R is quasi-free for GE2, (iii) R is universal for GE2, each implies the next, but as examples to be given later show, these three classes are distinct. 3. Direct and free products of GE-rings. We now turn to consider ring constructions which preserve the property of being a GE-ring. Here it is more convenient not to restrict the size of the matrices in any way. For any ring R and any integer n/> I we may define GEn(R), Dn(R ) and En(R ) as the subgroups of GLn(R) generated by all elementary matrices, all diagonal matrices and all Bq(a)(aeR, i#j) respectively. As in the case n=2, it is easily verified that EnD ~ = D~E n = GEn (R), and hence in every GE-ring, every AzGLn(R) has form (3. x ) A = DB(t)... B (r), where DeD, and B(PI=Bq(a) for some i#j, aER (depending on p). Let R be a direct product of a family of rings, say R=HRx, and denote the canonical projection R~R x by %. Then zx is a homomorphism which induces a 370 ON THE STRUCTURE OF THE GL~ OF A RING Ix group homomorphism ~x : GLn(R)-->GLn(Rx)" Composing these homomorphisms, we obtain a mapping F : GL~(R) ~ IIGLn (Rx) , which is easily seen to be an isomorphism. Now suppose that each R z is a GE-ring; then R need not be a GE-ring. We need only take a product of infinitely many factors R x such that in R z the integer r in (3. i) cannot be taken to be bounded, as X varies (1). Then it is easy to write down a matrix A~GLn(R ) which is not of the form (3. i). However, when the number of factors R x is finite and each is a GE-ring, then any AeGLn(R ) is again of the form (3. i) and we obtain Theorem (3. I). -- The direct product of a finite number of GE-rings is again a GE-ring. The result holds more generally for the direct sum (=weak direct product) of any number of GE-rings, since any matrix over the direct sum lies in the direct product of a finite number of factors. We next suppose that we are dealing with K-algebras, K being a commutative ring with I, and ask whether the GE-property is preserved under tensor products. The answer is easily seen to be " no ", even when K is a field, since K[x] is a GE-ring, for any indeterminate x over K, whereas the polynomial ring in two indeterminates, K[x,y] =K[x] | is not (see w 7). In fact the analogy with the Euclidean algorithm suggests that we consider, not the tensor product but the free product (cf. [5]). Of course it is now more natural to replace the K-algebras by K-rings, where K is any ring with I. It will be recalled that a K-ring is a ring R with a canonical homomorphism 0 : K---~R. In case 0 is injective, R is said to be a strict K-ring: thus a strict K-ring is just a ring in which K is embedded in a canonical way. In considering free products of rings we shall assume that all the factors are strict K-rings for a fixed K, which may thus be considered as a common subring of all the factors. To answer the question whether the GE-property is preserved by free products, would require a formidable calculation; we therefore confine ourselves to establishing the result in a special case (which turns out to be sufficient for many applications). Namely, we shall assume in addition that our rings are locally free ideal rings, i.e. semifirs (2). The GE-rings which are semifirs are characterized in the following Proposition (3-2). -- For any ring R, the following assertions are equivalent: (i) R is a GE-ring and a semifir, (ii) for any n >~ i, given al, ..., an, ba, ..., b,~R, if (3.2) Y~a~b i ~o, bl, ..., bn not all zero, then there exists CEEn(R ) such that (as, ..., an)C has at least one zero coordinate. (x) We shall see bclow (in w 5) that there even exist GEe-rings in which r cannot be taken bounded. (2) Cf. [6], where they are called local firs. This name has now been abandoned as it may give rise to confusion with local rings. Thus a semifir is an integral domain (not necessarily commutative) in which all finitely generated right ideals are free and any two bases of a given free module have the same cardinal. 871 x~ P.M. COHN A ring satisfying either (and hence both) of (i), (ii) will be described as a GE-semifir, for brevity. We note that H. Bass in [3] takes (ii) as the definition of a generalized Euclidean ring. Proof. -- Suppose (i) holds and (3.2) is given, then by Theorem 2.6 of [6] there exists AeGL,(R) such that (al, ..., a,)A has a zero coordinate. Since Ris a GE-ring, A-I=DE, where DED,, EeE n. Hence (al, ..., a,)E-1D -1 has a zero coordinate, and so does (al, ..., a,)E -1. Thus R satisfies (ii). Conversely, assume (ii) ; then R is a semifir, again by Theorem 2.6 of [6], so it remains to prove that GE, holds and here we may clearly assume that n> I. Let A be an invertible n � n matrix; if the first row has only one non-zero element then this must be a unit. By permuting the columns of A (which corresponds to right multipli- cation by a matrix in E,) we may bring this unit to the (I,I)-position and obtain :C o o o) * A 1 where AleGL,_I(R). Now the result follows by induction on n. If A has preci- sely r(> i) non-zero elements in the first row, we may assume that these elements come in the first r places. Writing A=(a0), A-l=(~ij.), we have, for j=2, ..., n, (S-S) i Zla. ~i = o. Since A -1 is invertible, either ~1i4:o for some j~>2 or a~i:~o for some j~>2. Choosing j accordingly, we may assume that of the % actually occurring in (3.3), not all vanish. By (ii) there exists BleEr(R ) such that (an, ..., al,)B 1 has a zero coordinate. Hence A(o has fewer than r non-zero elements in the first row and after at most n--i steps we reach the case r = I treated before. This completes the proof. In connexion with this proposition we remark that a GE-ring need not be a semifir. For by Theorem 3. I, a GE-ring need not even be an integral domain. To obtain an example of a GE-ring which is an integral domain but not a semifir, take any commu- tative local domain (i.e. an integral domain with a single maximal ideal) which is not a valuation ring. This is not a Bezout ring and hence not a semifir, but as we shall see later, any local ring is a GE-ring (w 4 below; see also [I2]). In the other direction we note that a semifir need not be a GE-ring; an example of a commutative principal ideal domain which is not a GE-ring will be given in w 6. We now show that the free product of any family of GE-semifirs over a skew field, 372 ON THE STRUCTURE OF THE GL 2 OF A RING is again a GE-semifir. Let R be any strict K-ring; by a suitable identification we may regard K as a subring of R. If K is a direct summand of R, as right K-module, R = K| then R is called an augmented K-ring, with augmentation module N. It was shown in [6] that for any ring K, the free product of any family of augmented K-rings exists and is again an augmented K-ring. We shall use the terminology and notation for free products introduced in [6]; in particular (H ") is the usual filtration of the free product (by height) and two elements a, b of the free product are said to interactif h(ab) <h(a) -}- h(b). In any ring R, two n-tuples (al, ..., an) and (ba, ..., b,) are said to be GE-equivalent if there is a matrix P=(p~i)eGE,(R) such that bj=Y_,aipli. Lemma (3.3). -- Let K be a field (possibly skew) and (Rx) a family of GE-semifirs. Denote the free product of the Rx (over K) by P and let (H") be the filtration by height of P. Let % b k (k = i, ..., r) be any elements of P such that h(akbk)-~ n and (3.4) ~akbk----o (mod H~-I). Further, assume that the a k are ordered by decreasing height, say h(ai)=m for i<<.s and h(aj)<m for j> s. Then either a) (al, ..., as) is GE-equivalent over K to an s-tuple (a't, ..., a's) such that h(a~)<~m with strict inequality for at least one i, or b) the elements a~ (i<~s) which interact with b i in a given factor R x are GE-equivalent over Rx to a tuple of elements of height <<. m, with strict inequality in at least one place, or c) for each a~ (i<<.s) there exist elements xqeP such that h(a~--Zaixi~)<m , h(a~xj~)<~m (j>s). The proof of this lemma will be omitted as it is very similar to that of Lemma 4- I of [6], the only difference being that instead of unimodular equivalence we assume and prove GE-equivalence throughout (1). As in [6], Theorem 4.2 we now deduce Theorem (3.4). -- Let K be a field (possibly skew) and (R~) a family of GE-semifirs. Then the free product of the Rz over K is again a GE-semifir. In particular, all the examples of semifirs given in [6] are actually GE-semifirs. For all arise as free products whose factors are either fields or rings of the form k[t] or k[t, t -1] and these are all clearly GE-rings. In particular, it follows that the group algebra over k of a free group (and the semigroup algebra of a free semigroup) are GE-semifirs, a fact used by H. Bass in [3]. (1) Note that the phrase " set including an element of height less than m " occurring twice in Lemma 4. x of [6] should each time be replaced by: " tuple of elements of height ~<rn, with strict inequality in at least one place ". Further, K should be a skew field. In fact this is what was proved (and used) there. 373 14 P.M. COHN 4- GL2(R) for a local ring. It is well known that any field (not necessarily commutative) is a GE-ring (cf. e.g. Dieudonn6 [8]) and this result was generalized to local rings by Klingenberg [I2]. In the present section we shall prove that a local ring is in fact a universal GE2-ring; however it does not have a unique standard form and in fact is not even quasi-free for GE~ (unless it is a field). To investigate the uniqueness of the form (4.I) [ct, ~]E(al)...E(a,)=I (~, ~eU(R), a#U0(R ) I<i<r), for any ring R, let us look at low values of r. Clearly, when r= o, we must have e=~=I. The case r=I is impossible, as we see by comparing (I,2)-elements. When r=2, we have eal=~a2=o, hence a l=a~=o and c~=~=--i. For r=3 a comparison of (2,2)-elements gives ~a~ = i, hence a 2 is a unit, which contradicts the conditions in (4. I). Thus in any ring R the relation (4. I) (with the stated conditions) is possible only if r 1> 4, apart from trivial cases. Next let r= 4 and for convenience write the relation as E(a~)F~(a~)F~(a3)E(a4) = [~, ~]. A comparison of terms shows that this holds if and only if ala~aaa4--alas--a3a4--ala 4 + I = o~, a 1 a s a a- a 1- a 3 -~ o, a s a 3 a 4- a s- a 4 = o, a 2 a 3 -- I = -- ~. We shall now show that in any local ring R which is not a field, these relations can be satisfied by elements ~, ~U(R), as, a3r ). For by hypothesis, U0(R ) 4= R; choose any elements a2, a 3 not in U0(R), then ~ = I--a~a3eU(R ) and al, a 4 are given by the equations al~+a3=o, ~a4+a2=o, which determine al, a 4 uniquely as elements of R; moreover it is clear that al, a 4 so defined do not lie in U0(R ). The fourth relation is of the form e ----- I + nonunit, and it determines e~U(R). Thus for r=4, (4. I) can be satisfied in any local ring not a field, hence such a ring is not quasi-free for GE~. Let R be a ring in which any one-sided inverse is two-sided and which satisfies the following condition: for any n> I, given al, ..., a~eR such that ~aiR= R, there exist b~, ..., b~_~R such that Y~(a~+a~bi)R=R. This is expressed by saying that n = I is a stable range for R; any such ring is easily seen to be a GE-ring (cf. H. Bass [2], who proves this and also shows that these rings include any ring which modulo itsJacobson radical satisfies the minimum condition on left ideals). The case n=2 of the above 874 ON THE STRUGTURt~ OF THE GL a OF A RING x5 condition states that for any a, beR such that aR+bR--R, there exists ceR such that a + beaU(R) ; clearly this holds in any local ring. Assuming this condition we can easily show that any AeGL2(R ) can be written in the form (2. i i) with r~< 3. For by hypothesis, for any B~GL2(R ) we can find ueR such that BE(u) -1 has a unit in the (I,2)-place and hence, for suitable wR, BE(u)-IE(v) -1 has 0 in the (I,2)-place. This matrix is therefore of the form Applying this argument with E(o)A in place of B, we find that E(o)A------ [e, ~]E(o)E(~-Ic)E(v)E(u), i.e. A =-- [~, e]E(~-%)E(v)E(u), which is of the required form. This shows in particular that any local ring is a GE2-ring. We now come to Theorem (4-x). -- Any local ring is a universal GE~-ring. After what has been said it only remains to show that any relation (4. i) in a local ring is a consequence of tlhe defining relations (2.2-4) and the relations in D,. Let (4.2) [oc, ~]E(al)... E(ar):I , ~, ~U(R), aiCUo(R ) for I<i<r. If i~>4, then as, a3r and hence as--I , a3--I~U(R), say a2---=y-[- i, a3=~+ I. By (2. Io), we can reduce (4.2) to --[~, ~]E(al--y-t)D(y)E(-- ~ --y-t--~-l)D(~)E(a4--~-t)E(as)... E(a,)= ~. Using (2.4) to pull D(y) and D(~) through to the left and then (2.9) (as in the proof of Theorem 2.2) we obtain an expression of the form (4.2) but with r replaced by a smaller value. This process can be continued as long as r/> 4; so we finally reduce (4.2) to the case r~< 3 and only (2.2-4) and their consequences have been used in the reduction. For r~< 3 we saw that the only relation (4.2) is --E(o)2=I and by (2.5) this is also a consequence of (2.2-4). Hence (4.2) itself is a consequence of (2.2-4) (and relations in D), as we wished to show. 5- Discretely normed rings. We now look for more general conditions on our ring R to ensure that the form (2.ii) for the elements of GE2(R), given by Proposition 2.2 is unique. The most natural way of imposing such conditions is by means of a norm; this is of course closely related to the question of the existence of a Euclidean algorithm, or a weak algorithm (cf. [5]) in R and hence to the question whether R is a Euclidean ring. In fact we shall use this method to construct rings which are not Euclidean in a rather strong sense: they are not GE-rings. 375 16 P.M. CIOHN Definition. -- A norm on R is a mapping I } from R to the nonnegative real numbers such that N. I. Ix[=o /fand0nly if x=o, N. 2. [x+y[<<.lxl+ly ], N. 3. lxy[ =lxl [Yl. We note that by N. 3, R must be an integral domain (not necessarily commutative). We shall say that I I is a discrete norm or that R is discretely normed, if further, N. 4. ]x]/> r for all x+o, with equality only if xeU(R), N. 5. there exists no xeR such that I<]xl<2. If 0~ is any unit, then by N. 4, ]e[>l ~, ]~162 ~> i, while by N. 3, hence I~[=I for all ~eU(R). Now N. 5 shows that [xl>~2 for any xCU0(R). Conversely, the two conditions N- 4' l~f=~ for an ~UCR), N. 5' I x]>12 for all xCU0(R ) imply N. 4 and N. 5- In order to derive further consequences of N. I- 5 we need an expression for the product E(al)... E(a,); this is closely related to the chain of equations in the Euclidean algorithm, cf. w 8. Let tt, t~, ... be any noncommuting indeterminates and define a sequence of polynomials in the t's with integer coefficients recursively by the equations e 1=0 , no= I~ (5"x) [e,(q, ..., t,)=e,_l(tl, ..., t,_l)t,--e,_z(tx, ..., t,_2). We note that for n i> o, the suffix of e, just indicates the number of arguments and so may be omitted when the arguments are given explicitly; we shall do so in what follows and only write the suffix when the arguments are omitted. We assert that _(e(aD...,a,) e(a~, . .., a,_l) ] (5.2) ( ~ " ., a,) --e(a2, ., at_l) ]" This is clear for r~ I and in the general case follows by induction, since, writing e~=e(at, ..., a~), e~ =-e(az, ..., a~+l) , we have ~--- r n 9 --e,_l From the symmetry of (5.2) it is easy to see that the e's may also be defined by e_t=o , no=I, e,(q, ..., t,~)= tle,_l(t 2, ..., t,)--e,_~(t3, ..., t,). The following lemma is basic for all uniqueness questions in GE2(R ) where R is discretely normed. E'aa)'"E'a"--\--e(a2,( ON THE STRUCTURE OF THE GL, OF A RING 17 Lemma (5- x). -- Let at, ..., a, (r> I) be any elements of a discretely normed ring R such that ai$Uo(R ) for i>I. Then (5.3) {e(ai, ..., ar)] >t {e(al, ..., a,_,) I and (5.4) e(al, ..., at) =~ O. If, moreover, the norm on R is such that (5.5) for any u eR, [ u ] = i and {I ~-U [ = 2 imply u =i, then the inequality in (5.3) is strict, unless 121 =2 and at=~eU(R), a2=a 4 ..... 2~ -1 and a3 = an ..... 2e. In this case, E(al)... E(a~)equals (: :) or (: :> according as r is even or odd. Then for i> I, {a~{ 1> 2, hence Proof. -- Write e~ = e(a t, ..., a~) for short. lei[ = {ei_tai--e~_2[ >1 2 {ei_t[-- [e~_2[, i.e. (5.6) le, l--le~-t[ ~> [e,-tl--{e~-2l (i> i). Suppose first that at=o; then el=al=o , e2=ata~--I =--I and hence by (5.6), ] erl--ler_l] i> le2l-lell = I. This proves (5.3) (with strict inequality) and hence (5.4)- Now let al+ o, then {a 11 >/ I and (5.7) [e~]--{eo] = [a~l-- I >~ o. Combining this with (5.6) we see that o for i>i, and hence (5.8) [er] >/le,_~{ t> .../> ]eli t> I. This proves (5-3) and (5-4). If further, lal[> i, then we have strict inequality in (5-7) and hence in (5-3). Likewise, if }al}> 2 for some i> i, then we have strict inequality in the corresponding formula (5.6) and hence in (5.3). There remains the case [at{ = I, {a~] = 2 (i> I); in particular, a~ is then a unit. Suppose first that at= I and assume that (5-5) holds. If we have equality in (5.3), then [ala~--I{=I, so ala2=i+b, where [b[=I, lI+b[=[ala2[=2 , hence b=I anda2=ataz=2. Now In general, when a x = and by induction it follows that all the al (i> i) must be 2. is any unit, we have I]A= [~-', I]E(~)E(a2)... E(ar) = E(I)E(a2~)E(~-la3)E(a,~)... 3 18 P. M. GOHN with a factor D = [I, g-l] (for r odd) or [~-t, I] (for r even) at the right-hand end, as follows by repeated application of (2.4)- Now if the elements of the first row of A have the same norm, then the same holds for [~-1, I]AD-1, so we are reduced to the case at= I and the result follows. Theorem (5.2). ~ Any discretely normed ring is quasi-free for GE2, and hence is universal for GE2. Proof. -- By Theorem 2.2, any relation in GE2(R ) can be brought to standard form [=, ~]E(at)... E(a,)= I, (5.9) for i <i<r. Comparing the (i,2)-elements in this equation we see where a~Oo(R ) that e(al, . .., at-l)=% and this contradicts Lemma 5- i if r-- I > i, i.e. r> 2. But when r~< 2, only the trivial cases noted in w 4 are possible. This completes the proof. Corollary. ~ A discretely normed ring which is a GE2-r/ng is a universal GE2-ring. When R is a discretely normed ring, it is easy to obtain an explicit form for the involutions in GE2(R), i.e. the elements of order two. This will be useful later on, in the study of automorphisms of GE~(R) and also in showing that for certain Dedekind rings the class number must be I. We begin by determining the centre of GE2(R), where R may be any ring: Proposition (5.3). -- For any ring R(4: o) the centralizer of E2(R) in GL2(R) consists of all matrices XI, where X runs over the central units of R. Proof.-- Let A=( a ~) ~GL2(R) centralize E2(R) ; since A commutes with E(o) we must have a=d, b +c=o; since it commutes with E(I) we have b =% i.e. A=aI and now the commutativity with E(x) (x~R) shows that a lies in the centre of R. This completes the proof. We next determine how the standard form simplifies in the case of elements of finite order. Proposition (5.4). -- Let R be a ring which is quasi-free for GE 2; then any matrix in GE2(R) which is of finite order modulo the centre of GE2(R) is conjugate (under E2(R)) to an element of the form (5. or (5" II) [~, [3]E(o)E(b), and in case (5. IO) holds, we have acUo(R ). 878 ON THE STRUCTURE OF THE GL 2 OF A RING x 9 Proof. -- Let A~GE2(R ) be of finite order modulo the centre of GE2(R ) and write (5. x2) A = [~, 13]E(a~)... E(a~). Then any conjugate of A is again of finite order mod the centre. If r>~ 3 in (5. I2) and alEU0(R), then A is conjugate to [9, ~]E(a')E(al)... E(a,-1), for some a'eR, and now the length can be reduced. Likewise, if areU0(R), we can transfer the factor E(al) to the right-hand end and then make a reduction. Thus we obtain a form (5-12) for a conjugate of A, in which either r~<2 or a#U0(R ) for all i. Since A'*=XI, we have [~, ~]E(al)... E(ar) [a, ~]E(al) ......... E(ar) =XI. By pulling all the diagonal factors through to the left and then transferring them to the right-hand side, we obtain a relation E(bl)... E(b,n ) = diagonal matrix, where each b i is associated to some a~. By hypothesis this can only hold if some bfiU0(R), hence some a~eU0(R ) and so in particular, r~<2. For r= I, we obtain the form (5. io), where neces,~arily aleU0(R ). If r----2, we may without loss of gene- rality assume that aleU0(R). Suppose first that al=TeU(R). Then transforming A by E(x), we have E(x)AE(x) -1 =E(x)[~, ~]E(y)E(b)E(o)E(--x)E(o) =--[~, ~]E(~-lx~--y-1)D(y)E(b--y-l--x)E(o). Choose x=b--y -~, then the right-hand side becomes and this is again of the form (5. I o). In the alternative case, a 1 = o and we obtain (5. i i). Finally if in (5.12) r=o, then this is of the form (5.II), with b=o. We remark that (5-IO) is conjugate, under GE~(R), to a matrix of the form [I, ~]E(a) ; for we can reduce c~ to I by transforming by [0r I]. It is now easy to derive a standard form for the involutions in GE2(R), when R is quasi-free for GE 2. Theorem (5.5). -- Let R be an integral domain which is quasi-free for GE2, then any invo- lution ~:--I in GE~(R) is conjugate to one of the form where h runs over a transversal of 2R in R (qua additive group). In particular, if 2R=R, then every non-central involution is conjugate to [i,- I]. 379 20 P.M. GOHN Proof. -- By Prop. 5-4 and the remark following it, any involution A+--I is conjugate to one of the forms (_; ;) In the first case, by squaring, we see that [~, ~]E(~-la)E(a)=I, hence a=o, ~=--i and we find A=[I, --I]E(o). Transforming by [I, --I]E(I)E(o) -1 we reach (5-I3) with h=I. In the second case squaring shows that ~2=~=I, hence e,~=4-i (because R is an integral domain) and ~(u + ~)b = o. Thus either -- ~ = e = 4- i and we have the form (5.i3) or ~=~=4-i and b=o. But this means that A= and these possibilities were excluded. Thus every involution can be brought to the form (5. I3). Now transforming (5. I3) by B~l(c ) we obtain 4- I and by a suitable choice of c we can ensure that h 1 = h-- 2 c belongs to the given trans- versal of 2R in R. In particular when 2 R = R this means that every involution is conjugate to :k [i, -- i]; since [-- i, I] is transformed to [I, -- I] by E(o), we need only one of these forms and the proof is complete. A ring is said to be 2-torsion free if 2 x = o implies x = o. With this definition we note the following application: Proposition (5.6). -- Let R be a 2-torsion free integral domain which is a quasi-free GE~-ring. Then every projective module on two generators is free. Proof. -- Any projective module M on two generators is a direct summand of R 2 and may be characterized by its projection, an idempotent 2 � 2 matrix, E say. The matrix P=I--2E has square I; if P=4-I, then E=o orI, andMis clearly free. Otherwise P is an involution *--I and by Theorem 5.5, after applying a suitable inner automorphism, we have o) In particular h -- o (mod 2 R), hence by a further inner automorphism we may arrange that P=4-[I,--I]; then E=[i,o] or [o, I] and M is again free. In a Dedekind domain each ideal is projective and can be generated by two elements; moreover, the free ideals are just the principal ideals, hence we obtain the Corollary. -- Let R be a 2-torsion free discretely normed Dedekind ring. If R is a GE2-ring, then it must be a principal ideal domain. In the next section we shall see how the result may be applied to purely imaginary quadratic number fields. We conclude this section by giving some examples of discretely normed rings. i. The rational integers, with the usual absolute value. More generally, consider the ring of algebraic integers in any imaginary quadratic number field. If this is 380 ON THE STRUCTURE OF THE GL 2 OF A RING 2! embedded in the complex numbers, the usual absolute value is a norm, i.e. it satisfies N. 1- 3. N. 4 is also satisfied because the integers form a lattice which can only have finitely many points in the unit circle; if any were strictly inside the unit circle, their powers would give an infinite set. Further, if an element is on the unit circle, so is its conjugate and hence the: element is a unit. Thus N. 4 always holds; N. 5 holds with a few exceptions (see w 6). Whenever it is satisfied we have a discretely normed ring. 2. Let R be any ring such that U0(R) is a field, k say, and suppose that we have a degree-function on R, i.e. to each a sR there corresponds a non-negative integer or --0% written d(a), such that D. I. d(a)=--oo if and only if a=o, D. 2. d(a)=o if and only if aEU(R), D. 3. d(a--b)<~max{d(a), d(b)}, D. 4. d(ab) -= d(a) § d(b). Then R becomes a discretely normed ring if we put l al = 2 a(~ For example, the polynomial rings over k, in any number of indeterminates, and the free associative algebras (polynomial rings in non-commuting indeterminates) are of this form. 3. If R is any discretely normed ring, then R[x], the polynomial ring in a single indeterminate over R, may be discretely normed by the rule l Y'a,x'l = Y la, i 2'. 4. The free product of fields does not at first sight admit a norm, but it does have a filtration which is very nearly a norm, and many of the results proved here for discretely normed rings can be carried over for free products of fields. We shall not enter into the details here (cf. [5])- In a discretely normed ring it is often possible to decide whether the ring is a GE2-ring. If it happens to be Euclidean with respect to the norm, it is clearly a GE~-ring. In the next two sections we shall show that under certain conditions a discretely normed ring cannot be a GE2-ring; this means in particular that no norm function can be defined for which the ring is Euclidean. 6. Rings of algebraic integers and algebraic functions. We have just seen that the ring of integers in an imaginary quadratic number field is discretely normed, provided that N. 5 holds. This amounts to the condition that no integer a in the ring satisfies I<N(a)<4. It is easily verified that this holds for the integers in Q(~v/~----d), where d is positive and squarefree, except when (6.x) d=I, 2, 3, 7, II. 381 22 P.M. COHN As it happens, these are just the fields in which there is a Euclidean algorithm with respect to the usual norm [a[=N(a) 112 (cf. [9], ch. 14). For the remaining cases we have Theorem (6. x), -- Let d be a square free positive integer and I the ring of integers in Q (~-~--d). Then I is not a GE~-ring, unless d has one of the values (6. I). Proof. -- We shall assume that I is a GE2-ring for some d which is not among the values (6. I) and derive a contradiction. By what has been said, I is discretely normed by [a[=N(a) 1/2, and since I is also a 2-torsion free Dedekind ring, it is a principal ideal domain, by Proposition 5- 6 Corollary. By Theorem 2.2, every element AeGL2(I ) has the form (6.2) A=[~, [~]E(qt)...E(q,), where [q~]1> 2 for i<i<r. Since [ ] is the ordinary distance in the Euclidean plane, condition (5- 5) of Lemma 5- i is always satisfied. Moreover, the only units in I are -4- I ; this is easily checked because the low values (6. i) of d have been excluded. Let us denote the first row of A by (a, b) ; then three cases are possible, according to the value of qr. (i) [q,[ 1> 2. Then by Lemma 5. I, [aI>[b[, unless a = (recall that Q-I are the only units in I). (ii) ]qr]=o and hence q,=o. Applying Lemma 5.I to AE(o)-!, we see that [a[<] b 1, unless a =+b. (iii) ]qr] = I and hence qr =-4-I. In this case we find that (6.3) ]a~bl<lbl for at least one choice of sign, unless a-4-b=b. Again this follows by considering AE(q,) -~. Now suppose that [a]=[bl, but a:#-4-b. Then the first two alternatives do not apply and a:~o, 2b, so that (6.3) must hold for at least one choice of sign. Thus if we can find a and b, forming the first row of an invertible matrix, such that (6.4) lal=lbl, [a+bl>~lbl, we have a contradiction. Now in a principal ideal domain any pair of elements without a common factor forms the first row of an invertible matrix, and any element of the field of fractions can be written as a quotient of two elements without a common factor. So to satisfy (6.4) we only have to find an element 0~EQ(%/~) such that (6.5) lOCI=I, [0r I; for on writing 0~ = a/b in reduced form, we obtain a solution of (6.4). Now the equa- tion [ 0~] = i defines a circle in R ~ (relative to oblique coordinates) on which the rational points are clearly dense. These correspond to points c~eQ(~/~) and by going suffi- ciently far from the real axis we can ensure that the second relation in (6.5) is also satisfied (for both signs). This gives us the required pair of values (a, b) and it shows that I is not a GE2-ring. 882 ON THE STRUCTURE OF THE GL~ OF A RING The first value of d not listed in (6. i) for which I is a principal ideal domain is d=i 9 (cf. [9], P. 213) and in this case an explicit matrix in GL2(I ) but not in GE~(I) is 2+o 3--0 5--20]' where 03--0-{-5 =o. By going through the steps of the proof of Proposition 5.6, Corollary, we can construct a similar example for Q(~/'~-5). In the language of valuation theory we can say that for a ring of integers in an algebraic number field to be discretely normed it is necessary for the field to have a single place at infinity. When this condition holds we can prove an analogue of Theorem 6. I for function fields (1). Theorem (6.2). -- Let K be afield of functions of a single variable with k as fMd of constants and assume that k has characteristic not two. Let p be any place of K/k of degree fp> I and denote by I the ring of elements of K which are integral everywhere except possibly at p. Then I is not a GE2-ring. Proof. -- If v~ denotes the exponential valuation at p then the product formula shows that vp(x)~<o for all xEI, x~eo, with equality only if x~k. If we define ~(x) =--v~ (x), then ~ is easily verified to be a degree function on I. Hence I is discretely normed and since it is 2-torsion free and Dedekind, we see that if I is a GE2-ring, it must be a principal ideal domain. Further, every AEGL2(I ) has the form A=(ca ~:) = [~, ~]E(ql)... E(q,) ~(q,)>o (z<i<r). As in the proof of Theorem 6. z there are three possibilities: (i) 3(fr)>o: then ~(a)>~(b) oi a=Xb (X~k), (ii) qr=o: then 3(a)<8(b) or a=Xb (Xek), (iii) 3(q,)=o: then q,;=Xek and S(a--Xb)<S(b). To avoid these three possibilities we need only find a pair of elements a, b in I without common factor such that S(a--Xb)=~(b) for all X~k. Let k(p) be the residue field at p and x-+x the residue mapping. By hypothesis, k(p)~e k, so there exists ~ek(p), ~r choosey~K suchthat ~=~. Then ~(y--X)=o forall X~k and as in the proof of Theorem 6. I we can write y = a/b, whence ~(a--Xb) = 3(b), so that none of (i)-(iii) can hold. Hence I cannot be a GE2-ring. When K is of genus zero, the condition of Theorem 6.2 is necessary and sufficient, for in that case .f~ = I implies that K is rational and I is just a polynomial ring over k. (1) I am indebted to J. V. Armitage for a helpful discussion of this point. 883 P. M. COHN ~4 7. k-rlngs with a degree function. We now turn to look at the second example of discretely normed rings given in w 5- Let k be a field (possibly skew), then a k-ring with a degree function was defined in w 5; we stress that an element of degree zero must by definition lie in k. In w 5 we showed that such a ring is discretely normed. We shall now show that for these rings we can strengthen Theorem 5.2 by proving that the standard form is unique. Theorem (7- I). -- Let R be a k-ring with a degree function (k afield); then R has a unique standard form for GE 2 . Proof. -- By Theorem 2.2 every AeGE2(R ) can be brought to the form (7. I ) A= [~, ~]E(al)... E(ar) (ai~R , o~, ~@U(R)), where a iek (I <i<r) and al, a s are not both zero in case r = 2. Suppose that we also have (7.2) A=[y, ~]E(bl)... E(b,) bi6k (I<j<s); we must show that ~=y,~=3 and r=s,a~:b~. We may assume r~>s and then use induction on r; further, we may assume that -(=~= I, by replacing A by [y, ~]-IA if necessary. If a, = b~ we can cancel the last factor and use induction, so we may assume that a,+b,. Then I--+[,, ~]E(al)... E(ar)E(o)E(--bs)... E(--bl)E(o ) = :t:[~, ~]E(al)... E(ar--b,)E(--b~_~)... E(--ba)E(o ). If we put a,--b~=c and multiply by E(o) we get (7.3) E(o) =DE(a1)... E(a,_~)E(c)E(--b~_a)... E(--bl) , where D is some diagonal matrix. Comparing elements in the (~,2)-place we find (7"4) e(a2, ..., at_l, c,--b,_l, ...,--b~)=o. We now apply Lemma 5. I and note that condition (5.5) is satisfied because for R the hypothesis is vacuous, and [ 21~< I ; therefore the inequality in (5.3) is always strict. Assume first that s~>3, then also r~> 3 and we reach a contradiction unless cek. But then c must be a unit because it is not zero, and (7-3) takes the form E(o) =DE(a~)... E(a,_l--c-1)D(c)E(--bs_t--c-~)E(--b~_2)... E(--bl). By hypothesis, at_a, b,_lek, hence ar_l--c -1, b,_l--c-lek. If we pull D(c) through to the left (by (2.4)) and equate the (2,2)-elements, we obtain e(a;, ..., a;_2, a'~_~--c, --b~_l--c -1, --b,_ 2, ...,--be)----o, where a~ is associated to a i and hence none of the arguments lie in k; but this contradicts Lemma 5- I. 384 ON THE STRUCTURE OF THE GL, OF A RING When s=2 we have bib2-- I bI1)=[~, ~]E(al). . . E(at), --b~ hence e(a2, ..., a,_l) .... ~-1. By Lemma 5. i, r-- 2 = O and in this case the uniqueness follows easily. When s = i, E(b) = [~, ~]E(al)... E(a,), hence e(a2,...,ar_l)=o. ByLemma5.i , r--2=--I, so r=i and the uniqueness is then clear. The same argument applies when s----o, and this completes the proof. We remark that the result does not hold for all discretely normed rings. E.g., for the ring Z of rational integers, In the next section we shall introduce a different generating set for GL2(Z ) in terms of which a unique normal form is possible. We now give some more specific examples of rings with a degree-function which are GE-rings and some which are not. Naturally, any Euclidean ring (with respect to its degree function) is a GE-ring, and for this to hold the ring need not be commutative, e.g. the ring k [x] of polynomials in a single indeterminate over a skew field is Euclidean. More generally, we have the rings with a weak algorithm described in [5]- We shall not recall the definition here but merely note that they are always k-rings with a degree function, for some field k. A typical example is the tensor ring over an arbitrary k-bimodule; in particular, any free associative algebra over a commutative field is of this form. For these rings we have Theorem (7.2). -- Any ring with a weak algorithm is a GE-ring. Pro@ -- By Prop. 3.2 we need only show: for any n~>I, given al, ..., a,, bl, ..., b, ~R such that (7.5) ~ga~b~=o b~ not all zero, then there exists CEEn(R ) such that (al, ..., a,)C has at least one zero coordinate. The proof is by induction on }]d(ai). By suitably renumbering the a's and b's we may assume that d(aibi)=m for i<~s and d(aibi)<m for j>s. Then (7.5)shows that al,..., a s are right R-dependent, hence by the weak algorithm, an a s of maximal degree, say a 1 is right R-dependent on the rest: a 1 = ~aic i -~ a; d(a;)< d(al) , d(ai61)(d(al). This shows that the transformation from (al, ..., a,,) to (a'~, a2, ..., an) is a product of elementary transformations and it diminishes the value of the sum ]~d(ai). Hence the result follows by induction. 385 26 P.M. GOHN Since a ring with a weak algorithm is discretely normed, such a ring is a universal GE~-ring; it is also easily seen that E,(R) is normal in GLn(R ). Next, to give some examples of rings which are not GE-rings, we establish a neces- sary condition for a k-ring with a degree function to be a GE2-ring. Let R be any k-ring with a degree function which is also a GE2-ring. We saw that in this case (5.5) always holds and 121~ I, therefore Lemma 5-I shows that in the present case d(e(aD ..., ar) ) > d(e(al, ..., at_l)) for any elements al,..., a, (r>i) such that a~(~k for i>I. Let us call a pair of elements (a, b) a regular row if it can occur as the first row of an invertible matrix over R (when R is commutative this reduces to the notion of a unimodular row defined by Bass in [2]). Take any regular row (a, b) and let A~GL2(R ) be a matrix in which it occurs as first row. Since R is a GE2-ring, A can be written in the form (7. i) and as in the proof of Theorem 6.2, there are there possibilities: (i) if q,r then d(a)>d(b), (ii) if q,=o then d(a)<d(b), (iii) if q~=~eU(R), then d(bo:--a)<d(b). In terms of the notion of (right) R-dependence, which is defined in any ring with a degree function (cf. [4] or [5]), we can sum up the result as follows: Proposition (7-3)- -- If R is a k-ring with a degree function which is also a GE2-ring , then of any two elements of the same degree which form a regular row, each is R-dependent on the other. With the help of this result it is easy to show e.g. that the ring k[x,y] of polynomials in two indeterminates over a field is not a GE2-ring. As the degree function we take the total degree in x andy. Then (I -t- xy, x ~) is a regular row consisting of two elements of the same degree, neither of which is R-dependent on the other. This means that the matrix ( i -~ xy x ~ ) __y2 i --xy which is clearly invertible, cannot be expressed as a product of elementary matrices. The same reasoning shows that more generally, the ring k [Xl, ..., xe] is not a GE2-ring whenever d>I. 8. Discretely ordered rings. The results of w167 5-7 show the usefulness of a norm for the study of GE2(R ). In some respects the same purpose is served by assuming that the ring is totally ordered (1); here it is again necessary to make some discreteness assumption. Definition. -- A discretely ordered ring is a ring R which is totally ordered, such that (8.x) for any a~R, if a>o, then a>~i. (1) I am indebted to H. Bass for drawing my attention to this possibility. 386 ON THE STRUCTURE OF THE GL 2 OF A RING In order to describe GE2(R ) in this case it is advantageous to supplement the diagonal matrices by matrices with positive coefficients and we therefore replace the generators E(x) by (8.2) P(x)=( x Io). Of course the P's could also have been used to develop the theory for discretely normed rings; but the E's there offered the advantage that their determinant was i. If P2(R) denotes the group generated by all the P's, then in particular, [I,--I]=P(I)P(--I)P(I)eP~(R), and since [I,--I] normalizes E2(R) and (8.3) V(x) = [I,- I]E(x), E(x) = [I,- I]P(x), it follows that E2(R) is a subgroup of index 2 in P2(R). In order to describe P2nD 2 we shall write (8.4) C (~) = [., --.- 1] (~ EU(R)). With these notations we have the following analogues of (2.2-4): (8.5) P(x +y) = V(x)P(o)P(y), (8.6) P(.)P(--~-I)P(.) = C(e), (8.7) P(x) [~, ~] = [~, .]P(~-lx.). Now the relation (8.3) between P(x) and E(x) shows that the defining rela- tions (8.5-7) are equivalent to the relations (~. 2-4). Hence R is universal for GE 2 if and only ifGE2 has (8.5-7) and the relations in D, as a complete set of defining relations. As in w 2 we obtain the following relations as a consequence of (8.5"7) : (8.8) P(o)' =I, [P(I)P(-- i)]3=--I, (8.9) V(x)-I = P(o)P(--x)V(o)---- ( ~ _;), (8. zo) P(x) V(y) -1- P(x--y) P(o), (8. z, ) P(x)P(y)-IP(z) = P(x--y + z), (8. x2) P(x) V(e) P(y) ~-P(x -5 .-i)C(-)P(y + e-~). In particular, putting ~ = I in the last relation, we find P(x)V(z)P(y) ---- P(x + i)C(I)P(y + I) ---- C(-- I)P(--x-- i)V(y + i), whence on replacing x, y by x--i, y--z respectively, we find P(--x)P(y) = (3(-- I)V(x-- I)V(1)P(y-- I). (8. ,3) Similarly, P(x)P(--y)P(z) =--P(x-- I)P(I)P(y-- 2)P(I)P(z-- I). (8.z4) Next we shall derive expressions for a product of P's, analogous to (5.2); they 387 28 P. M. COHN are essentially the continuant polynomials, cf. [i8] and [5]. Let tl, t2,.., be sequence of noneommuting indeterminates and define the polynomials p~ recursively b P_l=o, P0=I, (8. x5) p~(tl, ..., tn) =pn_l(tx, ..., t,_t)t ~ +p~_~(tl, ..., t,_~). As in the case of the e's defined in (5- r) we shall omit either the arguments or the suffix, from the p's. It is easily seen that Pla, } - "'" at) p(al, ...,ar_l)] P(al)... \p(a2, ., a,) p(a2, ., a~_l) ]" The symmetry of (8. i6) leads to the following alternative definition for the p's: (8. /P-l=o, P0= , t P,(tl, ..., t,)= t~p,_~(t2, ..., t,)+p,_2(ta, ..., t,). Either definition shows that p, may be described as the sum of t lt2 . . . t n and all tern obtained by omitting one or more pairs of adjacent factors t~t~+t. In particular, th shows the truth of Lemma (8.x.) -- In any ordered ring R, given al, ..., arER such that a~>o J I<<.i<~r (r>o), then p(al, ..., a~)>o. The same conclusion holds if al >~o , and ai>o for 2<<.i<<.r, provided that r >~ 2. This lemma leads to the following analogue of Theorem ~. 2 and Theorem 5- which is related to the uniqueness of the expansion of a rational number in a simp continued fraction (of. [9], P- I35)- Theorem (8.2.) -- Let R be any discretely ordered ring. Then R is universal for GE moreover, any A~GE2(R ) is unique of the form (8. x8) A = [,, ~]P(aa)... V(a,) a~eR, ,, ~eU(R subject to the conditions al/>O , ai~o (I<i<~ (8.x9) (8.20) when r = 2, a 1, a2 are not both zero. Proof. -- By definition, GE2(R ) is generated by the diagonal matrices and all P(c a~R; using (8.9) and (8.7) we can bring any such product to the form (8.i8). ai=o for some i (I<i<r) we can use (8.5) to simplify the expression (8.I8); so may assume that ai4=o (I<i<r). To complete the proof we use induction on t] number of sign changes in the sequence (al,..., a,_l, I). If this number is zer it means that the conditions (8. i9) are satisfied. In the contrary case let i be the 1~ suffix for which ai<o. Then --ai~o, hence --ai--I~>O and A = [0~, ~]P(al)... P(a,_I)C (- i)P(--a,-- ~)P(~)P(a,+I-- I)... P(a,) = ~=[,, --~]P(--al). 9 9 P(--a~_l)P(--a~--I)P(I)P(ai+,-- I)... P(a,). If a~=--I, we can reduce the number of factors P by (8.5). 388 ON THE STRUCTURE OF THE GL~ OF A RING 2 9 If ai+l> I, then there is one less sign change than before and we can apply the induction hypothesis. If a~+t=I , we can combine P(I) with P(ai+~) by (8.5) , unless i~-I =r; in any case the number of sign changes is again diminished by i. There remains the case ai+l<I ; this can only happen when ai+l~<o , hence i+I=r and the same conclusion applies. By induction it follows that the conditions (8. I9) can always be satisfied; moreover only (8.5-7) were used in the process. To prove that the form (8. I8) is unique, let us first assume that A is diagonal. If al>o , then a compa- rison of (I,2)-elements shows that p(al, ..., ar_l)=o , which contradicts Lemma 8. i. If a 1=o, we have P(a~)... P(a,) = DP(o), where D is a diagonal matrix, and comparing (2,2)-elements we find that p(a3, ..., a,_l)=o, which again contradicts Lemma 8. i, unless r~<2. In the latter case we are only left with the possibility a l=a s=o, which was excluded in (8.~o). If al<o, then r= i and A cannot be diagonal. Thus (8. I8) is unique when A is diagonal. In general, let (8. I8) hold and also A=[y,~]P(bl)...P(bs) bl>~o, bi>o (I<j<s). If a,~..-=b~ we can cancel a term and use induction on max(r, s), so we may assume that ar~e bs, say a,.>b~. Writing D1, D2, etc., for diagonal matrices whose exact value is immaterial, we find (8.2I) I : D1P(al)... P(a,.--b~)P(--b,_l)... P(--bl)P(o), hence I = D~P(--al)... P(b,--a,.)P(bs_l)... P(b2)P(b~--I)P(I)P(-- i) = D3P(al)... P(ar--b,-- ~)P(I)P(b~_~-- I)P(b~_2) 9 9 9 P(b2)P(b~-- I)P(I)P(-- I). Since a,.>b~, we have ar--b~--I~>o ; in case this is zero we can combine P(a,_l) with P(I). Then all arguments are strictly positive except possibly the first, last and b1--I. Moreover, a~>o, b~>o. Assume for the moment chat b~>o, then bl--I ~>o and by combining P(b2) with P(i) in case b1--I =o, we obtain a formula in which all arguments except the first and last are >o, and the first is />o. The special case proved shows that this is only possible when there are just two arguments, both zero. Thus we have uniqueness in this case. If b 1 =o, the argument is the same except that the last two factors in (8.eI) are omitted. This completes the proof. The most obvious example of a discretely ordered ring is the ring Z of rational integers. Unfortunately this cannot be extended to the ring of integers in a real algebraic number field, since such a ring is never discretely ordered. If R is any discretely ordered ring, then R[x], the ring of polynomials in an indeterminate over R, is again discretely ordered, if we take as positive polynomials those polynomials which have a positive 389 3 ~ P.M. COHN leading coefficient. E.g. Z [x] is discretely ordered in this way. We shall show below that this ring is not a GE2-ring , but in order to do so we need a refinement of Lemma 8. I. Lemma (8.3). -- Let R be discretely ordered, at, ..., a~eR (r>~ 2), al>~ o, a~> o (1 <i<r) ; then (8.22) p(al, ..., ar)>P(al, ..., a,_l). Proof. -- Write pi=p(al, ..., a~) and assume first that a14: o; then by definition, p, =p,_la, +p~_~, hence Now a,>o, hence ar~> I and pr_x>o, p~_2>o by Lemma 8.1, hence P~>Pr-1, i.e. (8.2e) holds. If al=o , then by (8.I7) , p~=p(a3, ..., a~),p~_l=p(a a, ...,a~_t) and the result still holds. Now let AeGEz(R), where R is any discretely ordered ring. Then by Theorem 8.2, A = [e, ~]P(ql) 9 9 9 P(q,), ql>~o, qi>o (i<i<r). (8.23) Let A=(: ~) and consider c--dqJ" If we compare this expression with (8.23) we see by Lemma 8. i, that b and a--bq, must have the same sign; replacing A by -- A if necessary, we may take this sign to be positive. Then b>a--bq,; now there are three possibilities, according to the sign of qr: Lemma (8.4). -- Let R be a discretely ordered ring, and let AeGE2(R ) be any matrix with first row (a, b), where b > o and assume that A has the form (8.23) with r >1 2. Then (i) /f q~>o, then a>b>o, (ii) /f q,=o, then b>a>o, (iii) /f qr=--c<o, then b>a>--bc. Proof. -- A comparison of (I,2)-elements in (8.23) shows that b =~p(al, ..., at_l) , hence e>o and so ~ may be ignored. If q,>o, we can apply Lemma 8.3 to the first row of A. If q, = o we apply Lemma 8.3 to AP(q~)-t unless r~< 3, when a separate argument is necessary; this may be left to the reader. If q~<o, q~=--c, we have b>a+bc>o and hence the result follows. As an application, let us take any discretely ordered ring R and consider R[x], with the ordering by highest coefficient described earlier. The matrix {I +2x 4 A=~ --x 2 I--2x 390 ON THE STRUCTURE OF THE GL 2 OF A RING 3x is invertible and I +2x>4>o , so if AeGEe, we are in case (i) of Lemma 8. 4 (clearly r>/2 in any representation (8.23)of A). But then q~>o and b>a--bq,>o, i.e. a> bq,> o. Let u be the leading coefficient of q,; then u>o, hence u~> I and so 4u>o, 4u>2; this shows that q, must be of degree zero in x. But for such q~ we have a--bq,>b, which is a contradiction. Hence A cannot be of the form (8.23) and this shows that R [x] cannot be a GEe-ring. In particular, Z[Xl,..., xd] is not a GEe-ring, for any number (>o) of indeterminates. Similarly, free associative rings (i.e. algebras over Z) are not GEe-rings. 9" The commutator quotient structure of GE2(R) and of E~(R). For any group G we shall denote the derived group by G', the commutator quotient group G/G' by G a (i.e. G abelianized) and write x-+x ~ for the natural homomorphism G-+G a. We saw in w 2 that for any ring R, E2(R ) is normal in GE2(R ). When R is universal for GE2, this quotient is abelian; we therefore begin by determining the quotient in this case. Theorem (9. x). -- Let R be any ring universal for GE2. Then (9. x ) GEe(R )/Ee(R ) -~ U(R)t Proof. -- We define a homomorphism f: GE2(R) --> U(R) ~ by the rule E(x) -+ To show that this is well-defined we need only check that the defining relations (2.2-4) are preserved. But this is immediate for (2.2) and (2.3), and for (2.4) it follows from the fact that (c~[~)a=(~,) ". Clearly f maps GE2(R ) onto U(R) a, so to complete the proof we need only show that kerf=Ee(R ). By definition, E(x)ekerf, so the kernel contains Ee(R ). Conversely, if A, given by (2. i i) say, lies in the kernel, then 0r eU(R)'; multiplying by D(~)=E(o)eE(~)E(~-I)E(~), we may suppose that ~ = I, so it only remains to show that [y, I]eE2(R) for all -feU(R)', and this will follow if we prove that [~-a[~-i~{3, I]~E2(R ) for all ~, [3eU(R). Now we saw that D(~)eEe(R ) and (9.2) [~-i~-1~}, I] = D(~-I)D(~-I)D(=})eE2(R ). Hence kerf--Ee(R) and (9.1) follows. Corollary 1. -- In any ring which is universal for GE~, D~(R) aE2(R ) is generated by all matrices D(~), 0cell(R). For if we restrict the homomorphism f: GE2(R ) -+ li(R) ~ constructed in the proof of the theorem to De(R), the kernel is just Da(R) nE2(R ). Now [% ~] maps to i if and only if ,~ eli(R)', but in this case the proof of the theorem shows that [0c, ~] can be written as a product of matrices D(y), yell(R). 391 32 P.M. COHN Cor. I shows that any element AeEy(R ) can be written A----D(~I)D(~2)... D(%)E(al)... E(ar). In particular, any relation in Ey(R ) can be brought to this form, using only (2.2), (2.3) and (9.3) E(x) = which is just the special case of (2.4) where ~ = I. If R is assumed to be quasi-free for GEy, then by the remark just made, we can further reduce this relation to the form (9.4) D(~I) 9 9 9 D(~,) ---- I. Using (9.2) we can finally reduce (9.4) to the form (9-5) [Y1, i]... [y,, I]:I where yieU(R)', yi...y,:i. Thus we obtain Corollary 2. -- In a ring R which is quasi-free for GEy, Ey(R ) is generated by all E(x) (xeR) and if D(e) (eeU(R)) and [y, I] (yEU(R)') are defined by (2.3) and (9.2) respectively, then a complete set of defining relations in terms of these generators is given by E(x) E (o) E (y) = -- E(x+y), E(x)D(c~) = D(~-I)E(~xc~), [Y1, I]... [Yl, I] = I (YieU(R) ', Y1... Y~-~ i). In the case of universal GEy-rings, Theorem 9. i may be thought of as a generaliza- tion (in the case n = 2) of Dieudonnd's determinants over a skew field (cf. [7]). Since the right-hand side of (9. i) is abelian, we obtain Corollary 3. -- For any ring which is universal for GEy, (9.6) Ey(R) GEy(R)'. This corollary can still be improved, as follows: Proposition (9.2). -- For any ring which is universal for GE~ and in which 1 can be written as the sum of two units, (9.7) GE~(R)'-~ E~(R), and hence GEy(R)a~U(R) a. The result follows, once (9.7) is proved, and this is well known (cf. e.g. [2]). It is proved by noting that if a+~---- i, the commutator X-tY-1XY of X-=--[~ -1, I] and Y----E(~-tx)E(o) -t is E(x)E(o)-l; by transposition we obtain E(o)-lE(x) and hence GEy(R )' contains --n(o)n(-- I)n(-- i)E(o)E(o)E(x-- I)= E(o)E(o)D(--,)E(x) = E(x). In particular, when U0(R ) is a field k, we have the Corollary. -- In any k-ring with a degree function, (9.7) holds provided that k has more than two elements. 392 ON THE STRUCTURE OF THE GL z OF A RING It is convenient to consider next the group Ez(R) a. Let R be any discretely normed ring; working mod E2(R )' we have by (2.5) and (2.2), E(x)E(y) = E(x)E(o)E(y)E(o)E(-- I)E(o)E(-- I)E(o)E(-- i) = E(x +Y--3), hence (9.8) E(x)E(y) = E(x +Y--3) (mod E2(R )'). This suggests defining a mapping E2(R)-+R by putting (9.9) E(x)-+x--3. This mapping can be extended to a homomorphism of E~(R) into the additive group of R provided that the defining relations of E2(R ) are preserved. Since E(o)2=--I, we must have --I-+--6 and in particular, (9. IO) I2=o in R. Next the definition of D(0:) shows that (9.xx) D(0:) -+ 2c~ A- ~-1--3, and using this value in (9.3) we find that x-- 3 -4- 2 ~ + 0:- 1__ 3 = 2 0:- 1 + 0:__ 3 + ~x0:-- 3, i.e. ex0:--x=0:--a -1. If we replace x by x+~ -t, this equation reduces to (9. ,2) .x~ = x. Thus in order to be able to define a homomorphism into the additive group of R we must have (9. to) and (9.12). However we can obtain a homomorphism in all cases if we divide out by the appropriate subgroup of R. In order not to complicate the result, let us assume that U(R) is abelian. Then we have Theorem (9.3). -- Let R be a ring which is quasi-free for GE 2 and such that U(R) is abelian, and denote by M the additive subgroup of R generated by 12, all o:xot--x (xeR, c~eU(R)) and all 3(0:-f-i)($-4-i) (0:, ~zU(R)); then (9. I3) E2(R)~ ~ R/M, under the homomorphism defined by (9" I4) E(x) -->x-- 3 (mod M). It should he noted that M is in general not an ideal of R but merely an additive subgroup. Proof. -- It is clear from the remarks preceding the theorem and the definition of M that the first two relations of Theorem 9. I, Cor. 2 are preserved. Moreover, the third relation reduces in the present case to (9-IS) D(0:- 1)D(~- 1)D(~) = I. 5 34 P. M. COHN By definition of M we have ~xo~ - x, hence e-- ~-~ (mod M). Thus D(~) -+2c~ + ~-~--3 --- 3( c~- I) - 3(~ -1- I) (mod M), and the left-hand side of (9- 15) is mapped to 3(o~-- I) -~- 3(~-- I) -~- 3(0~-- I) ~ 3(~ @-~ @-o~--3) - o (mod M). Thus all defining relations of E~(R) are preserved and (9.14) defines indeed a homo- morphism g : Ee(R ) --~R/M. Clearly g is onto, and since the right-hand side is abelian, ker g~E~(R)'; to complete the proof we must establish equality here. By (9.8) we have E(x + 3)E(y + 3) --- E(x +y + 3) (mod E~(R)'), (9.x6) and hence E(xl + 3)..- E(x, + 3) --- E (Y~x~ + 3) (mod E~(R)'). (9.17) If the left-hand side maps to zero, so must the right-hand side, whence ]~xi-= o (mod M). Now we have the identity E(x)E(o)-IE(x)-'F~(o) = E(x)E(o)-~ F~(--x)= -- F,(x) E(--x) ; hence --E(x)E(--x) - I (mod E~(R)'). It follows that D(e ~) = -- D(e)D(-- 0c) = E(~)E(c~-~)E(c~). E(-- ~)E (-- e-~)E(--~) -I (mod P.2 (R)'). Next we have hence E(x)D(e) = D(~-I) E(~x~), E(c~x~)E(x)-l--D(~ ~) --I (mod E2(R )'). (9" 18) If we put x=--3,y=o in (9-I6), we see that E(3) eE2(R)', and so, by (2.8), we obtain E(~x~--x + 3) = E(~x~) E(x)-IE(3) ~E2(R)'. (9" I9 ) (9.18) we also have E(exc~)- E(x), hence E(~) = E(e -1) (mod Ez(R)'), and so By hence E(3~) = E(e)E(o)E(~)E(o)E(~,) -= E(o)~E(~)E(c~-t)E(~) -= D(~), (mod E2(R )'). (9.20) E(3~ ) -D(e) In particular, E(-- 6 + 3) = E(-- 3) = --I, whence (mod E~(R)'). (9.2x) E(I2 +3) =I Further, E(3e~)-D(c~)--D(c~)U(~)-E(3c~)E(3fi) and so E(3 (o~ ~ + o~ + ~+ i) + 3) -= E(3 ~)E(3 ~)E(3 ~)E(I 5) - D(c~)D(c~)D(~) - I, 394 ON THE STRUCTURE OF THE GL= OF A RING by (9.18) and (9.'~I), hence (9.2 o) E(3(~- ~ I)(~ -{- I) 2r-3) -I (mod E2(R)'). Now (9. I9), (9.2 I) and (9.22) show that for any generator x of M, E(x+3) eE=(R)'. By (9. I7) this holds for any xeM. Hence, in particular, if the left-hand side of (9.17) is mapped to zero by g, then the right-hand side lies in E=(R)' and therefore so does the left-hand side. This shows that ker g_cE=(R)' and it completes the proof of Theorem 9.3- The isomorphism (9.13) can be made to play a similar role for E2(R ) as the deter- minant for GL2(R); in this connexion we note that when R is commutative, so that a determinant is defined, then (in case R is discretely normed or discretely ordered) E2(R ) is just the subgroup of GE2(R ) consisting of matrices of determinant I, because every matrix in GE2(R ) is congruent (mod E2(R)) to a matrix of the form [~, 1]. In particular, for a commutative (discretely normed) GE2-ring R we have E=(R) = SL2(R ). As an example, take the ring Z of rational integers. Here M = I2Z and Theorem 9.3 shows that SLe(Z) ~ is cyclic of order i2. Secondly, let I be the ring of Gaussian integers. The only units are 4-i, 4-i and in this case M is generated by 12 and all 2 x and all 3(~ q- I)($ + I). This is just tlhe ideal 2 R, so SL2(I)" is the direct product of two cycles of order two. Suppose that R is a k-ring with a degree-function and moreover, that k is in the centre of R, i.e. that R is a k-algebra. Then M contains x(cr for all xeR, aek (~4=o), and hence coincides with R, unless every non-zero element of k has its square equal to I. In the latter case k cannot have more than 3 elements and then it is easily verified that M = o. Thus we have Corollary 1. -- Let R be a k-algebra with a degree-function, then E~(R)~= o unless k is the field of 2 or 8 elements, in which case E,(R)"=~R. At the other extreme R has no units apart from 4- i ; then it is easily seen that M is the additive subgroup generated by i 2. This is true in particular when R is discretely ordered, and so we have Corollary 2. -- Let R be a discretely ordered ring (or a discretely normed ring in which :k i are the only units), then E2(R)a~R/I 2. For example, if I is the ring of integers in an imaginary quadratic extension of Q, which is not Euclidean, then E2(I) a is the direct product of an infinite cycle and a cycle of order 12. To obtain more precise information about GE~(R) a it is necessary to find out more about E2(R)/GE2(R )' in the cases where Prop. 9.2 does not apply. Let us return to the proof of Theorem 9-3 but drop the assumption that U(R) is abelian. By (2.8), E(~(x--3))E(x--~)-*E(3~) = E(~(x--3)--(x-- 3) + 3~) 395 3 6 P. M. COHN shows that E(3 @ =D(~) (mod GE~(R)'); the proof uses no units apart Now (9" 2o) from ~ and so does not depend on the commutativity of U(R). Since we clearly also have I = E(x)-~[~, I]-XE(x) [~, I] - E(x)-~E(c~x)D(c~) (mod GE2(R ) '), we deduce that E((~-- I)X + 3) ---- I (mod GE~(R)'). (9.~'3) By symmetry we also have E(x(0~--i) ~- 3) -- I (mod GE~(R)'). (9. ~'4) Let N be the ideal of R generated by all c~--i (~eU(R)). We note that N~_M sincee, g., ~x~--x-~(e--i)xe+x(a--i) and I2=--6(--1--1). It is easily seen that the mapping h : E(x)-*x--I (mod N) p~eselves the first two relations of Theorem 9. i, Cor. 2, while the third one becomes trivial (mod GE2(R)'); therefore we have a homomorphism E2(R )/GE2(R )' -+R/N. If the left-hand side of (9. I7) lies in the kernel of this mapping it follows that ~xieN and now (9.23), (9.24) show that the right-hand side of (9-I7) must lie in GE2(R)'. This proves that the kernel is in fact equal to GE2(R )' and we have Theorem (9.4). -- Let R be any ring which is quasi-free for GE 2 and denote by N the ideal generated by all or--I (~eU(R)). Then Ir~ E2(R )/GE~(R) =R/N. E.g., when R=Z, then N=2Z and we obtain the result of Hua and Reiner [ii], that SL2(Z)/GL2(Z )' is cyclic of order two. If I is the ring of Gaussian integers, N=(~ +i)I and SL2(I)/GL~(I )' again is cyclic of order two. Theorems 9. I and 9.4 may be summari'zed as follows: Theorem (9.5). -- Let R be a ring which is quasi-free for GE 2 and denote by N the ideal generated by all ~--I (~EU(R)). Then there is a split exact sequence o -~ R/N -+ GE21'R) a ~ U(R) a -+ o. For the mapping e~-+[~, I] clearly induces a splitting. xo. Generating sets and free products in GE2(R). It is well known that SL2(Z), the two-dimensional unimodular group, can modulo its centre be written as a free product of a 2-cycle and a 3-cycle. It seems to be unusual for GE2(R ) or E~(R) to be expressible as a free product in a non-trivial way, but there are a number of results on large subgroups which have the form of free products (cf. e.g. Nagao [14] ). The question is related to the problem of deciding 396 ON THE STRUCTURE OF THE GL 2 OF A RING when GE2(R) is finitely generated; by means of the normal form for elements in GE2(R ) we shall be able to give an answer to these questions which includes many of the known results as special cases. We begin by discussing the problem of finite generation of GE2(R). Theorem (xo. i ). -- Let R be any ring; if U(R) is finitely generated (as multiplicative group) and R # finitely generated, as U(R)-bimodule, then GE2(R ) #finitely generated. Proof. -- Let U(R) be generated by 71,-.., 7h and R, as U(R)-bimodule, by c1,..., Ck. We already lmow that GE2(R) is generated by all E(a), a~R, and [~, ~], ~, ~EU(R). By (2.2) and (2.4), E(a) can be expressed in terms of E(c,), E(o) and [~, [~], and hence in terms of (io.x) E(o), E(c~) (r= ~, ..., k), [u yj] (i,j: i, ..., h). This shows that GE~(R) is finitely generated. The following is a partial converse: Theorem (xo.~). -- Let R be a ring quasi-free for Gig s and such that GE2(R ) is finitely generated. Then so is U(R) a, and moreover, R is finitely generated, as U(R)-bimodule. Proof. -- Since GE2(R ) is fnitely generated, so is U(R) a, by Theorem 9.5. Moreover, every generating set of GE2(R ) contains a finite subset which is still a gene- rating set. We can therefore find elements cl, ..., ckeR and Y1, -.-, YheU(R) such that the elements (Io. I) generate GI~.z(R ). Thus every element of GE2(R ) can be expressed in the form (xo.2) [~, ~]E(@... E(ar) , where al, ..., ar belong to the U(R)-bimodule generated by Ca, ..., ck. Clearly we may assume that a~4=o (i--~I ..... , r) and using (2.9) we may even take ad~U0(R), provided that we let the a~ range over the U(R)-bimodule generated by i, q, ..., c~. Now take any b~R and consider E(b). If E(b) is expressed in the form (lO.2), we necessarily have r~> i, hence we obtain -- [~, ~]E(al)... E(a~_l)E(a,--b)E(o ) = I. If a,.--br then the argument used to prove Theorem 5.2 shows that r~> 2 and we obtain a contradiction. Therefore ar--bsU0(R ) and it follows that b lies in the U(R)-bimodule generated by I, q, ..., ek; but b was any element of R. This shows that R is finitely generated, as U(R)-bimodule. This theorem shows e.g. that GL2(k[x]), for any field k, is not finitely generated (cf. Nagao [I4] ). It seems likely that Theorem IO. 2 holds for any ring which is uni- versal for GEe. Next we turn to the question of free subgroups and free pioducts. Let R be any ring, and A a subgroup of the additive group of R. Then it is clear that the set B12(A)-={B,2(a) ] aeA} 397 3 8 P. M. COHN is a subgroup of G2(R ) isomorphic to A, and of course the same holds for its conjugate with respect to P(o), which will be denoted by B~a(A ). Theorem (IO.3). -- Let R be a ring which is quasi-free for GE 2 and A, B any two subgroups of the additive group of R which do not meet U(R). Then the subgroup of GE2(R) generated by BI~(A ) and B~I(B ) ~ equal to their.free product. Proof. -- Let F be the subgroup generated by BI~(A ) and Bal(B ). Any element of F has the form f-glhlg2hz...g, hr, where g,~B12(A), h~eB~i(B) and &+I for i0eI, h~+I for i+r. Write g, = B12(a,) = E(ai)E(o) -1, h, = B21(b,) = E(o)-~E(b,), then f has the form (xo.3) zkE(aa)E(ba)... E(a~)E(br), where ai, bi:~o except possibly az or b~; moreover, ai, biCU(R), by hypothesis. In particular, the left-hand side of any relation in F can be brought to the form (io.3). But by hypothesis there are no non-trivial relations of the form (IO. 3) and it follows that F is a free product as asserted. For example, if x, y are elements of R such that nx, t~r ) for all n+ o, then the subgroup generated by Bxe(X ) and B~I(y) is free on these generators. In particular this yields the well known result that for any indeterminate x over a field of characteristic zero, Bx2(X ) and B21(x) generate a free group. Similarly, for any integer m>~ 2, Bxe(m) and B~x(m ) generate a free group. I I. The construction of homomorphisms between general linear groups. Let f: R->S be any homomorphism (a) of rings; clearly this induces a homo- morphism of n� matrix rings f~ : R~-->S~ and since a unit of R~ maps to a unit of S,, we obtain a homomorphism (II.I) f* : GL~(R) -+ GLn(S ). Secondly, let g : R--->S bc an antihomomorphism; this induces in the same way an antihomomorphism of groups g* : GL~(R) --> GL~(S). If we follow this by inversion in GL.(S) we again obtain a homomorphism of groups (I1.2) g: GL~(R) -+ GL,~(S). In particular, when S = R, the automorphisms and antiautomorphisms of P~ give rise in this way to automorphisms of GL~(R). Not all automorphisms arise in this way, however; if a is any homomorphism of GL~(R) into the group of central units of R, then the mapping A-->A.A ~ is an endomorphism of GL,(R), which wilI be called (~) It is understood that such a homomorphism maps the I of R to the i of S. 398 ON THE STRUCTURE OF THE GL 2 OF A RING a central homothety. When we come to study automorphisms of GL,(R) we shall find that for n 1> 3 and a fairly wide class of rings, every automorphism of GL,(R) is obtained by combining/* or ~ with an inner automorphism and a central homothety. For n = 2 the situation is rather different; to describe it we need a Definition. -- Let R, S be any rings. A U-homomorphism f:R-+S is a homo- morphism x~x' of the additive group of R into the additive group of S such that (II,3) I'=I, and (II "4) (,a~)'-=-o~'a'~' for all a~R, o:, ~eU(R). If g is a homomorphism from the additive group of R to that of S which satisfies (I I. 3) and instead of (I 1.4) satisfies (xx.5) (~a~)'=~'a',' for all a~R, v., ~U(R), then g is called a U-antihomomorphism. Clearly a U-homomorphism between fields is just an ordinary homomorphism, and likewise for antihomomorphisms. More generally, we have Proposition (xx.x). -- Let R be a ring which is generated, qua ring, by its units. Then a U-homomorphism of R into an arbitrary ring is a homomorphism and a U-antihomomorphism is an antihomomorphism. Pro@ -- Let x-+x' be: a U-homomorphism and write Ro=:{x~Rl(xa)'=x'a" for all a~R} then by (II,,4), R0 contains U(R) and if x, yeR o then for any aeR, [(x--y) a]'= (xa--ya)'= (xa)'-- (ya)' = x'a'--y'a' = (x'--y') a' = (x--y)'a', hence x--yeR o. Further, (xy)'~-x'y' and for any a~R, (xya) '-= x'(ya)'= x'y' a'= (xy)'a', which shows that xyeR o. Together with (I i. 3) this shows that R 0 is a subring of R and since it contains the units of R, by (II.4), it must be the whole of R. But this means that f:x-+x' is a homomorphism. The proof for antihomomorphisms is similar. Tile form of the defining relations (2.2-4) now yields almost immediately Theorem (ii .2). -- Let R be any ring universal for GI~.2, S any ring and let f: R-+S be any U-homomorphism x-+x'. Then 9 r induces a homomorphism s) /* : Gv. (R) -+ GE (S) by the rule (II.7) E(x)-+E(x'), [~, ~]--+ [~', ~']. 399 4 ~ P. M. COHN If g : R->S is a U-antihomomorphism, then g induces an antihomomorphism g* defined in the same way and hence induces a homomorphi~m (xx. 8) ~ : GE2(R ) --> GE2(S ) by the rule (II "9) E(X) "+ E(x') -1, [9~, ~] ---> [~', ~,]-1. Proof. -- We need only check that the defining relations (2.2-4) and the relations in D~(R) are satisfied and this is clearly the case. Whenf is a homomorphism, the effect off* in (i i. 6) can be described more simply by the rule ,,,,o, (: (:: but it should be noted that for arbitrary U-homomorphisms this is not so. Although there is still a group homomorphism 2'* (given by (II.7)) it does not agree with the mapping (i I. Io), which in general will not be a homomorphism. A similar remark applies to U-antihomomorphisms. Corollary. -- Any U-homomorphism of a universal GE~-ring R defines a homomorphism (i~ .6) of GL2(R ) and any U-antihomomorphism defines a homomorphism (I~ .8) of GL2(R ). When the units in R (together with o) form a field k, a U-homomorphism of R is just a k-bimodule homomorphism such that i -+ i. Thus it induces a monomorphism from k to the image ring, so a U-homomorphism in this case is essentially a k-semilinear mapping. Below we list some special cases of Theorem i i. 2. (i) Let R, S be k-rings and assume that R is a universal GE2-ring with U0(R ) =k. Then any k-semilinear mapping of R into S induces a homomorphism of GL2(R ) into GL2(S). (ii) If R is a k-ring with a weak algorithm, then any k-semilinear mapping to a k-ring S induces a homomorphism of GL~(R). (iii) Any k-linear mapping of a k-ring R with a weak algorithm (into itself) induces an endomorphism of GL2(R ). In the case where R is the ring of polynomials over a commutative field in a single indeterminate, this construction of automorphisms of GL2(R ) is due to Reiner [i5] , [I6]. As an application of the above results, consider the free associative algebra over a commutative field k, on a free generating set which is at most countable. As k-bimodule this is just a vector space of countable dimension over k and so is isomorphic to k[x], the ring of polynomials in a single indeterminate. Hence the GL~ over the free associa- tive algebra is isomorphic to GL2(k[x]). In case :t: I are the only units in R, the statement of Theorem I i . ~2 can be sfightly simplified. Theorem (II "3)" -- Let R be any ring which is universal .for GE2 and in which i are the only units, and let S be ary ring. Then any additive homomorphism Jrom R to S which maps 400 ON THE STRUCTURE OF THE GL 2 OF A RING 4 I I-+I induces a homomorphism of GE2(R ) into GE2(S ) by the rule E(x)---,E(x'), and another one by the rule E(x)-+E(x') -1. Corollary. -- Let R be any ring universal for GEz, bz which 9 I are the only units, then there is an automorphism of GE2(R ) defined by w(x) The statement of the wtrious special cases (in particular the case where R is a universal GE~-ring) may be left to the reader. The result may in particular be applied to the case of discretely ordered rings, where the hypothesis relative to the units in R is automatically satisfied. I2. The analysis of isomorphisms of general linear groups. We now consider the converse problem: Given an isomorphism between GLn(R ) and GLn(S), when is this induced by a mapping f : R-+S? The aim will be to show thatf can be taken to be an (anti)isomorphism or in case n = 2, a U-(anti)isomorphism. We shall limit ourselves to k-rings with a degree-function and to begin with we assume n= 2. .Just as for fields, the case of characteristic 2 has to be treated separately. On the other hand, we do not need to restrict ourselves to GE2-rings, but instead consider rings in which projective modules are free. Lemma (x2. x ). -- Let R be a k-ring with a degree function, where k is afield of characteristic not two, and assume that either of the following conditions is satisfied: (i) R is a GEz-ring, (ii) ever); projective (right) R-module on two generators" is free, then any pair of anticommuting involutions in GL2(R) can be transformed simultaneously by an inner automorphism to the forms [I,--I] and P(o)=(~ ;)respectively. Pro@ -- Let A, B be the given involutions. If (i) holds, then by Theorem 5.5, A is conjugate to [I,--I], because A is non-central. If (ii) holds, consider E= I/2(I+A); clearly E is an idempotent, and since A+ it follows that E4:o, I. If we regard E as acting on R ~, it defines a direct decomposition and by assumption (ii), both the kernel and image of E are free R-modules; taking a suitably adapted basis, we obtain E in diagonal form: E ----- (; o~ and in this coordinate system, A = [I, --, ]. Thus A has been transformed to the form [i,--i], assuming only (i) or (ii). Let B--(: ~), then A-1BA=(__: --~) and byhypothesisthisequals--B, hence a=d-----o. Moreover, B~=I, therefore bc=i, i.e. b, cEU(R) and transforming B by [b, I], which leaves A unchanged, we obtain [c, i]B[b, I]=P(o). Now let R be a k-ring and S a k'-ring, both with a degree-function, where k, k' are 401 42 P. M, COHN fields of characteristic 4= 2 and assume that one of R and S, say S, is either a GE2-ring or has its projective modules free. Let (I2. I) f : GL2(R ) -> GL2(S ) be an isomorphism, and for brevity write Do=[i , --I] (in R or S). Now over any integral domain, --I is characterized as the only central involution, hence (--I)f------I. Since D o and P(o) are a pair of anticommuting involutions in GL2(R), their images underf are again anticommuting involutions in GL2(S), and the same is true of :q(Dof), ~(P(o)J), where ~ is one of the numbers I, --I. Let us make a definite choice of ~, then by composing 3 c with an inner automorphism of GL2(S) we may assume (by Lemma i2. I) that (12.2) D0,f= ~qDo, P(o)f= rjP(o). By (I~.2) the centralizer of D o in GL2(R ) is mapped into the centralizer of D o in GL~(S). But this centralizer is easily seen to be the set of all diagonal matrices, thereforef maps the subgroup D2(R ) onto D~(S). Our next problem is to characterize /T TD_--D-1TD, the triangular matrices T(h)----[~. Writing T=T(h), we clearly \u hi)" have (x2.3) (TD0)2= I, TI)T = TT D for all DeD~(R). (x2.4) I, oHowst theim geU=T s ti . thes meequ tion D----[I, ~], we have UDU=UU D, and equating the (i, I)-terms in this equation we obtain 02.5) b(~--~-l)c= o. If we assume that k' has more than 3 elements, we can find 3~k' such that 324: i, and now (i 2.5) shows that b ---- o or c---- o. In the remaining case k' is the field of 3 elements (because the characteristic is 4: 2) and from (12.3) we obtain (UD0)2= I, whence by equating coefficients, (I2.6) ab=bd, ca=&, aZ--bc=d2--cb= I. Moreover, in this case, T3= I, hence U 3-= I; in detail U = ca + dc d2 -]- cb \ 2 dc 2 d2~ I 2d3 _d_C_2dcb , thus a(ea~--i-]-2bc)=i, d(2d~--i-]-2cb)=i. This shows that a, deU(S) c c_k', i.e. a, d= + i, therefore a 2 = d 2= i and the last equation (I2.6) now shows that bc = o. Again we deduce that either b = o or c = o. 402 ON THE STRUCTURE OF THE GL~ OF A RING Let us assume that b+o, say, then c=o, and by (i2.6) which always holds, we have a2----dZ=I, i.e. (a+I)(a--1)=o. Since S is an integral domain, a----4-1, and similarly d=4-i; moreover, the first equation (12.6) shows that a=d. This then shows that U is of the form and clearly every such matrix satisfies the conditions (12.3) and (12.4) for T. Simi- larlyif b=o but c+o we obtain the form and this matrix again satisfies the conditions (12.3) and (I2.4). Finally, if b=c=o, then all we can conclude is that a=4-I and d=4-I, and thus we have 4-I or 4-D 0. This exhausts all the possibilities. Now f leaves D O fixed except for a scalar factor and it maps diagonal matrices to diagonal matrices, hence/maps any matrix T(h) to a matrix of the form (I 2.7) or (i 2.8). Consider T(I); if T(I)J=T(k) say, where keS, then k+o because f is injective. But for any heR, T(h)f commutes with T(k)=T(i)f, and it must therefore be again of the form (12.7), so in this case the subgroup BI~(R) is mapped into the subgroup 4-B12(S ) byf. The other possibility is that T(I)=B21(k); in this case we modifyJ by composing it with the inner automorphism defined by E(o) = DoP(o ). This leaves (i 2.2) unaffected except to replace ~ by --~, but now T(I)J=E(o)-tB~l(k)E(o)=T(--k), so that this is essentially reduced to the previous case. More precisely, we have shown that by composing f with a suitable inner automorphism we can ensure that (12.2) holds (for ~=4-I) and B12(R ) is mapped into +BI~(S ). As a consequence we can write (x~,. 9) T (x)f= ~(x)T(x~ 4- i}. where x-~.x ~ is a mapping of R into S and x-+r is a mapping of R into Since T(x)T(y)=T(x+y), we have +y)T((x +y)") = ~ = o +ya), hence (x +y)o = x o +yO, (I2. IO) = +y). (X2.XX) Putting y=x in (12.Ii), we find that ~(2x)----~(x)~=1, hence ~(x)=t for all xeR. We saw that x~ for x+o, hence (12. to) shows that the mapping ~ is injective. Repeating the argument with of replaced by f-t (and reading (I2.2) in the opposite 403 44 P. M. COHN direction), we see that G is an isomorphism of the additive groups of R and S. Next we look at E(x): E(x) = DoT(x) P(o ) . Applying f, we obtain E(x)f= r~2DoT(x~ hence E(x)y= E (x~ If we applyf to (2.3) and use (I2. I2) to simplify the result, we get E((a.- t)~ E (0d) E ((0r 1)~ = -- D (0r 1)f. The right-hand side belongs to D2(S), hence so does the left-hand side; but this can only happen if oUeU0(S ). We know that 0r176 and so we find D(~-l)f=__E((~,-1)o__(eo)-~)D(0d)E((a-1)o (eo)-t). Equating the (i, 2)-terms we see that (I2.X3) (~-1)o =(~o)-~, so both sides of this equation may without risk of ambiguity be denoted by ~-~ Using (I2. I3) on the preceding equation we obtain D(~-I).f=D((~~ or replacing ~. by ~-1 and using (I2.I3) again, (12.,4) D(cc)f= D(~). In particular, taking ~= I, we obtain (I2. I5) I ~ I. We can now use Hua's Theorem (cf. e.g. [I], p. 37) and conclude from (I2. IO), (12. I3) and (12. I5) that a is either a homomorphism or an antihomomorphism of k into k'. The same argument applied to f-t shows that a is actually a bijection between and k', and therefore an isomorphism or an anti-isomorphism. If we apply f to (2.4) we obtain D(0~-~176176 0~ ~ -- E(x~ ~ = D (~- ~176 and hence (x2. I6) (~x~) o = ~oxo~ o (x~R, ~U(R)). This relation can actually be used in the proof of Hua's theorem; more generally, we can use it to show, in exactly the same way as in the proof of Hua's theorem, that we have either or (x2. x8) (~x~) ~ = ~~176 ~ for all x~R and ~, ~U(R). Thus e is either a U-isomorphism or a U-anti- isomorphism. Next we consider the effect off on diagonal matrices. Every diagonal matrix can 404 ON THE STRUCTURE OF THE GL~ OF A RING be reduced to the form [i, ~] by multiplying by a matrix D(~), for suitable ~. Now D(~)f is given by (i2. r4) ; we may therefore restrict our attention to [I, 0~]. We know that this is again mapped to a diagonal matrix by f, say (12. I9) [I, ~]f--- [~X, ~]. Since P(o) [I, a]P(o) = [e, ~]-- [e, ~-~] [~, ~], we have [0r ~ 0r -~ [0~ x, o~ ~] = P(o) [o~ x, od'] P (o)= [o~ ~, o~X], and hence (x2. ~,o) ~ = ~~ Using this relation we obtain i.e. [~, Nf=([~, ~-~] [i, ~N)f= b ~ ~-~ [(~)~, (~)o(~)x], (I2.2I) [~, ~]f= [o~o, o~-o(o~)o] (~)x. Suppose now that o is a U-isomorphism; then (I2.2I) states [~, ~]f=[~~ ~](~)~. If we apply this to the equation (i2.22) [~, 0r [I, 0~][i, ~], we obtain [I, 0d }~ (0r x = [I, od]0~z[I, ~3 ~ ~x i.e. (~)~= [i, ~-~ V]~ ~. Equating the (I, 1)-terms in this equation we find (~)~--~, and it can be used to simplify the equation between the (2, 2)-terms to (x2."3) ~o~x=~o. Thus X is a homomorphism of U(R) into U(S), and since ~ maps U(R) onto U(S), (12.23) shows that 0r x centralizes U(S). By (2.4) [~~ 0d] (~)XE((~-ax~) ~) =E(x ~ [~~ t3~ (~I3) x = [~~ ~~176176 ~. This shows again that (0~x~)~176176176 and taking ~-=I, x=yo~ -1 we find o~XE(y ~)=E(y)0r x, which shows that o~ x lies in the centre of S. Thus X is a central homothety, and (I2. i2), (i2.2i) show that f is just the isomorphism induced by the U-isomorphism a, followed by the central homothety X. Finally, a comparison of (i 2.2 I) and (I2.2) shows that ~=(--i) x. Next assume that ~ is a U-anti-isomorphism, then (I 2.2 i) becomes [0~, ~3]f= [0d, 0r176162 x. 405 46 P.M. GOHN If we express ~ in terms of ~r and ~ by means of (12.2o), we can write this relation as [~, p]f= [p-o, ~-o](~p)~. Applying this to (12.22) we find that ~ is a homomorphism of U(R) into the centre of U(S) and as before we can use (2.4) to show that ~ in fact lies iu the centre of S. Thus f is now the isomorphism induced by the U-anti-isomorphism ~, followed by the central homothety ~. The result may be summed up as Theorem (I2.2). -- Let R be a k-ring and S a k'-ring, both with a degree-function, where k and k' are any fields of characteristic 4:2 and S is either a GEz-ring or all projective S-modules on two generators are free. Then every isomorphism between GL2(R ) and GL2(S ) is obtained by taking the isomorphism induced by a U-isomorphism or U-anti-isomorphism, followed by a central homothety and an inner automorphism. This includes the result of Reiner [16] when S=R=k[x] and earlier results of Schreier-v.d. Waerden and Hua (cf. [8] and the references given there). IfR is any k-ring, then the characteristic ofk is =t= 2 if and only if GL~(R) contains a central involution. Hence if we are given an isomorphism f : GL2(R)~GL2(S), where R is a k-ring and S a k'-ring, then k, k' both have characteristic =~ 2 or both = 2. In order to deal with the latter case we nced a Lemma analogous to Lemma 12. i. Whereas Lemma 12. I shows that in characteristic not two (and under the given hypo- theses) the subgroups of GL2(R ) of the type of the dihedral group of order eight form a single conjugacy-class, the next lemma establishes a corresponding fact for subgroups of the type of the symmetric group of degree 3, in the case of characteristic 2. Lemma (x2.3). -- Let R be a k-ring with a degree-function, where k is a field of charac- teristic 2, and assume further that R is a GEz-ring. Then any pair of involutions in GL2(R ) whose product has order 3, can be transformed simultaneously by an inner automorphism to the forms T(I)(=BI~(I)) , P(o) respectively. Proof. -- Let A, BeGL2(R ) be the given involutions and write C=AB. By Prop. 5.4 and the remark following it, C can be brought to one of the forms (x2.24) [I, ~]E(a), [0~, ~]E(o)E(b). Suppose that C has the second of these forms; comparing terms in the equation C 3 = I, we see that ~3 = ~.~ = I, b0~ 2 + ~b~ + ~Zb = o. If we transform by E(~b~) -1, we therefore obtain E(~be) [,, ~]E(o)E(b)E(o)E(~'b00E(o)= [~, e]E(~be2)E(o)E(b)E(o)E(~ebo~)E(o) -- [~, ~]E(b+ ~Zb,+ ~b,2)E(o) Thus C has been transformed to diagonal form. If C has the first form in (i2.24) , a 3 thcequation C3=I showsthat [~=a 2, I. When a=~ I, it follows that a2+a+ I =o, 406 ON THE STRUCTURE OF THE GL,~ OF A RING 47 transforn ng by (i i) I ~ we reduce C to the form [~, I]. Thus C has been reduced to either of the following forms We assert that in fact C can always be reduced to the first of these forms. Since C has order 3, we have ,a=~a__ I and ,, ~ are not both i in the second form (i2.25). Supposethat-=I, sayandlet A=( a bd), thenbyequating(I,I)-termsintheequa- tions A2=(AC)2=I, we find aS+bc=a2+b~c=i, hence b(~--I)C=O; therefore b or c must vanish, say b=o. Then a----d--i, and equating (2,2)-terms in (AC)8=I we find ~2= I, i.e. ~---= I, which is a contradiction; the same reasoning applies if c=o. Thus neither :c nor ~ in (I2.25) can be I and they therefore satisfy the equation (x2.26) xS + x + I =o. Assume first that this equation has a root in the centre of k, say (0. Then (.--(0)(.--(08 ) =% hence .----(0 or .=(08, and likewise for ~. If ~=., then C is a scalar matrix; leaving this case aside for the moment, so as not to interrupt the argument, we may assume that ~4=~; it f~176 that ~--"-t, andtransf~ ~) we obtain the first form in (12.25) for C. If (I2.'26) has no roots in the centre Z of k, it must be irreducible over Z and so all its roots in k are conjugate (Herstein [io]). Hence there exists yek such that y-l~y = e-1 and transforming [,, ~] by [i, y] we are reduced to the previous case. Thus C has n~ been reduced t~ the f~ ( i I ;) (with the exception noted). Now by hypothesis, C=AB, C-I=BA, therefore (x2.27) CB = A----- BC -1 . Taking B=(; ~) and using the expression found for C we obtain by equating components in (12.27) and simplifying, z=u, u+v+w=o. IfweequatetermsinB2=I, wefind thatuv=vu, u2+vw=I. Moreover, A=(: ;); and by Theorem 5.5, A is conjugate to T(h), for some heR. Thus there exists an inver- tible matrix P=(c a b) such that (I ~ . 28) AP = PT(h). Equating coefficients, we obtain the equations va + wc = a, vb + wd = ah + b, ua + vc = c, ub + vd = bh + d, 407 48 P.M. COHN or after some simplification, ( v -1- i ) b + wd = ah, (I2.29) i (V'~- I)a=ZOC, ub+(v+ I)d=bh. Now (12.28) may be written as (A+I/P=P( ~ ~). The first row of this matrix equation reads (I2.30) (V-~- I, w)P=(o, ah), hence (I2.31) (/) -~- I, W)=(O, ah)P -t. From the first two equations (i2.29) , we have (x2.32) (v+ I)C +w(a+c)=a, and by (i 2.3 I), ah is a common left factor of v + I and w, therefore a = ahk, whick shows h to be a unit, h=-h say. Combining (12.32) and the third equation of (12.29) we obtain an equation which in matrix form can be written Using (i2.3o), we may write this as ,,,-1[ b+c~ (o, If a=o, then c=t=o and the first two equations (12.29) show that w=o, v==i, anc hence u= i. In this case transformation by C -1 achieves the desired reduction If a~= o, then a~ + o, and from the last equation written it follows that p-l{ b+c~q for keR. \ d + (a + c)~qJ = t o ] some Hence d+(a+c)~]= \o]' Inserting these values in P, we find that thus b=c~q+ak, d=(a+c)'~+ck. ak + c~q ~ c I k p [ a c ~ to be invertible, evidently it commutes witt This shows the matrix 1=/c\ a+c C=(: ;) and transforming a by PI we obtain T(x ). The same transformatior reduces B=AC to the form P(o), as we wished to show. 408 ON THE STRUCTURE OF THE GL z OF A RING It only remains to show that the case C =coI, where co is a root of (I2.26) in the centre of k, cannot occur. Let us assume the contrary and write R- ={x Rlxco =co x}. For brevity the elements in R + will be called symmetric and the elements in R- skew. By hypothesis, every element of k is symmetric; further, every element of R can be expressed as the sum of a symmetric element and a skew element in just one way, for if (I=.33) x=x +@x- (x~t, z=~), then (I2.34) cox=x + co q- x-co ~, and solving the equations (12.33-4) we find x + ----- xco 2 -~ cox, x- ----- xco -[- cox; thus only one choice is possible for x +, x- and this choice clearly satisfies (I2.33). By (i2.27) we have coB = Bco 2, hence all the elements of B are skew. Now B is invertible and hence of the form [~, ~]E(ql)... E(q,). Denoting the first row of B by' (a, b), we have a = bq~ + a', (I2.35) where d(a')<d(b), by Lemma 5-i. If we equate the symmetric components here we obtain b(qr)- +(a') + =o. But d((a ' ) + )<~d(a ' )<d(b), hence (a') + =(qr)--=O and so a' is skew and q~ is symmetric. Therefore BE(qr) -1 consists entirely of skew elements; by induction on r we conclude that [~, ~] consists of skew elements, but this contradicts the fact that k_c R +. This completes the proof of Lemma ~2.3. With the help of this lemma it is now an easy matter to obtain an analogue of Theorem I2.2 for the case of characteristic 2. Let f: GL2(R) ~ GL2(S ) be an isomorphism, where R is a k-ring and S is a k'-ring, both with a degree-function, k and k' are fields of characteristic 2 and moreover S is a GE~-ring. In GL2(R ) the matrices T(I), P(o) form a pair of involutions whose product has order 3, hence so do their images in GL2(S), and by Lemma I2.3 we may therefore assume (by combiningf with a suitable inner automorphism of GL2(S)), (i2.36) T(I)f= T(I), P(o)f= P(o). 409 50 P.M. GOHN The subgroup B12(R ) of GL~(R) may be characterized as the maximal abelian subgroup of exponent 2 in the centralizer ofT(i) in GL2(R ) (1). It is therefore mapped to B12(S), the maximal abelian subgroup of exponent 2 in the centralizer of T(I) in GL2(S ). Thus there is a mapping a : R-~S such that T(x)f= T(x ~ xsR. By (2.2), we have (x +y) ~ = x ~ +y~ (I2.37) and by (I2.36), i~ I. (x2.38) Now a diagonal matrix D in GL2(R ) is characterized by the fact that both D and P(o)DP(o) normalize B12(R). This shows that f maps diagonal matrices over R to diagonal matrices over S; we can now follow the proof of Theorem i2.2 exactly and finally obtain Theorem (x2.4). -- Let R be a k-ring and S a k'-ring, both with a degree-function, where k, k' are any fields of characteristic 2 and S is a GE2-ring. Then every isomorphism between GL2(R ) and GL2(S ) is obtained by taking the isomorphism induced by a U-isomorphism or a U-anti- isomorphism, followed by a central homothety and an inner automorphism. In conclusion we briefly discuss the isomorphisms of GL,(R). It turns out that by a method similar to that used in proving Theorem I2.2, we obtain Theorem (i2.5). -- Let R be a k-ring and S a F-ring, both with a degree-function, where k and k' are fields of characteristic ~ 2, and assume further that everr finitely generated projective S-module is free. Then every isomorphism between GL,(R) and GL,(S) (for n >~ 3) is obtained by taking an isomorphism or anti-isomorphism.from R to S, followed by a central homothety and an hmer automorphism. Proof. -- GL,(R) contains a system J of 2" commuting involutions, namely [~: I, ..., :t: i]; moreover, the symmetric group Z of degree n acts on this set by permu- tations: there are n+I orbits, of (;)elements respectively (k=o, i, ..., n). The isomorphism f transforms J into a set of 2" commuting involutions in GL,(S). Now if P is any involution in GLn(S), then E -. 1/2(1-1-P) is an idempotent, so that we get a set of 2 ~ commuting idempotents in the matrix ring S,. Since projective S-modules are free, any idempotent can be diagonalized, and likewise any commuting set of idem- potents can be transformed simultaneously to diagonal form, with diagonal elements :5 I. By applying this transformation we thus bring the 2 n commuting idempotents to the form E=[el,...,e,], where e i=o or i. Now P=2E--I is again an involution, so that we have a system of 2" commuting involutions in diagonal form in GL,(S). More precisely, by applying a suitable inner automorphism, we may assume that for any (a) This subgroup may also be described as the set of all involutions in the centralizer of T(I), together with I. 410 ON THE STRUCTURE OF THE GL 2 OF A RING 5x involution P=[~I, "" ", gn], *i=-t-I, 1)j iS again diagonal, with on the diagonal. Ifk of the ~i are +i and the rest are --i, P is said to be of type (k,n--k) ora (k, n--k)-involution. The subgroup of X centralizing a (k, n -- k) -involution has the form Xk� hence ifP is of type (k, n--k), then P/is of type (k, n--k) or (n--k, k), i.e. either P/ or --Pf is of type (k,n--k). Consider the n (i, n--I)-involutions P1, ..-, P,; they form an orbit under Z, hence P1/, --., P,J are conjugate and so they are all of type (1, n--i) or (n--l, I). By combining/with a suitable inner automorphism we may thus assume that (I2.39) PJ=~PI (~ =+:, i= I, ..., n). Since every involution in fir is a product of P~'s, we have, for any involution PGJ, P/=~pP where ~e=~q k if P has type (k, n--k). Let So be the permutation matrix corresponding to the permutation ~eX; then S~-IPiSo = Pio, hence applying / and noting (I2.39), we obtain (So/)-*P, (So/) = P,o. Therefore So(So/) -* commutes with Pi for i= i, ..., n and hence is diagonal, say (12.4o) So/=DoSo. Consider the n-cycle --(1 2 ... n) and let Dp=[%, ...,%]; then theequa- don S~=I shows that (I2.4I) O~nO~n_ t . . . (Xl= I. If we transform the images under / by a fixed diagonal matrix T = ['(1, 9 9 Y,], the PGJ remain unchanged and (12.4 o) takes on the form Sa/= D'o So, where D'~ = T-1Do. S~TS2 a. In particular, we have D'p = [u 7~-1%71, ..., 7~-1%`(,_1]. Our aim will be to choose ~==kI and Yi(i=I, ...,n--I) such that (I2.42) y~-1~iy~_l= ~ (i-= I, ..., n; go-----Ya= I). If this is to hold, then by (12.41) we must have yl=~l~, ~'2=0~2y1~---r '1, ..,, 7n_l=~n_l... IXl~ n-t, yn=~ n= I; conversely, these equations will ensure that (12.42) holds. Now these equations can always be solved if n is even, while for odd n they can be solved provided that ~ = I. To sum up, we can always transform by an inner automorphism (induced by a diagonal matrix) such that for p=(r ~ ... n), (:2.4 ~ ) reduces to ( 2.4a) so/=z(p)so, where X is a linear character of Y,. 411 52 P.M. COHN Next consider a transposition, say z=(I2); if Dr=[~l, ..., [5,], the relation 2 2 St-- I shows that ~1~2-- I, ~i = I (i/> 3). If we put Sr into diagonal form for a moment we see that it is a (I, n--i)-involution, and hence Srfis of type (i, n--i) or (n--i, i); this means ~3 = ~4 .... -- ~5, (-- + i). Thus D,=[~, ~-1, 8, ..., 8] (~=+i). Now the equation (i 2 ... n)=(n n--i)(n--i n--2) ... (3 2)(3 I) shows that s0 = "-3... st, where S p denotes the transform of S by Sp. Applying J and equating diagonal terms, we find that If n is odd, Z(P) = I and these equations reduce to In this case D r = 3I is a scalar matrix and since ~, -r generate Z, every D o is scalar, in fact, 02.45) soy= Z(~)So, where Z(~) is a linear character of Z, the identity or the alternating character according as 8=I or =--i. Ifnis even, the equations (12.44) reduce to (= and in this case ~ is still at our disposal. We now choose ~--= Z(P); then D r is again a scalar matrix and so (12.45) holds in this case too. Now the diagonal matrices may be characterized as the set of matrices centralized by the P's; consider the set centralized by 02.46) ..., Clearly this is the set of all A~-[d3, ..., d,], where AeGL2(R). Hence the set centralized by the elements (I2.4 6) and the permutations not involving I or 2 is A@XI,_ 2 (Zek). ThusJ'induces a mapping A~-I,_~ ~ Af~-X(A)I,_e, wherefis an isomorphism GL2(R)-§ GL2(S). Transforming both sides by S o (for ~eX) we see that a corresponding formula holds for other rows and columns. Now by Theorem 12.2, fis of the form fl Jeff, where fl is induced by a U-isomorphism or a U-anti-isomorphism ?, say the former, f2 is a central homothety and f3 is an inner automorphism. Thus f~ maps Bij(x ) to B0.(x~), and since (B~z(x), B23(Y)) = B13(xy ) (where (A, B)=A-~B-~AB), we have (B~2(x~), Bea(y~)) = Bla((xy)~), i.e. 412 ON THE STRUCTURE OF THE GL 2 OF A RING Thus ? is in fact an isomorphism; similarly a U-anti-isomorphism is shows to be an anti- isomorphism. By taking commutators with elements of the form I, ~-B we see that k(A) lies in the centre of S; dividing by ;~(A) we thus have a homomorphism Aq- In_~ ~ 0(A)A~ q- In_2, and by definition, 0(A) = I on involutions. Further, [0r I, I, ..., I] ---> [0(0r ~, 0(0r I, ..., I]; permuting the second and third rows and columns (which does not change the left- hand side) we see that 0([~., I])= I. Therefore 0 = I and the result follows. REFERENCES [i] E. ARTIN, Ge0m~tr/c Algebra (New York, x957). [2] H. BAss, K-theory and stable algebra, Publ. math. LH.E.S., n ~ 22, Paris (1964). [3] H. BAss, Projective modules over free groups are free, 07. of Algebra, I (I964) , 367-373 . [4] P- M. COH~, On a generalization of the Euclidean algorithm, Proc. Cambridge Phil. Soc., 57 (I96Q, 18"3o- [5] P. M. COHt% Rings with a weak algorithm, Trans. Am~r. Math. Soc., Io9 (I963), 332-356. [6] P. M. COHN, Free ideal rings, J. of Algebra, i (1964), 47-69. [7] J. DIEUDONN~, Les ddterminants sur un corps non commutatif, Bull. Soc. Math. France, 71 (I943) , 27-45. [8] J. DmUDONN~, La gdomdtrie des groupes classiques, Ergeb. d. Math., n o 5 (Berlin, I955). [9] G. H. HARDY et E. M. WRIGrIT, Introduction to the theory of numbers, 2nd ed. (Oxford, I945). [io] I. N. HERSTEm, An elementary proof of a theorem of Jaeobson, Duke Math. 07., 2I (I954), 45-48. [ I I ] L. K. HUA et I. REINER, Automorphisms of the unimodular group, Trans. Amer. Math. Soe., 71 ( 1951 ), 331-348. [I2] W. KLINOENBERG, Die Struktur der linearen Gruppen fiber einem nichtkommutativen lokalen Ring, Archly d. Math., 13 (I962), 73-8L [i3] J. LANDIN et I. R~Im~R, Automorphisms of the two-dimensional general linear group over a Euclidean ring, Proc. Amer. Math. Soc., 9 (I958), 2~ [i4] H. NAOAO, On GL(~, K[x]), J. Inst. Polytech. Osaka City Univ., Ser. A, Io (I959), II7-I2I. [I5] L REINER, A theorem on continued fractions, Proc. Amer. Math. Soc., 8 (I957), IIII-III 3. [I6] I. P, mINER, A new type of automorphism of the general linear group over a ring, Ann. of Math., 66 (i957) , 461-466. [I7] B. L. v. d. WAERDEN, Moderne Algebra, I (Berlin, i937). [18] J. M. H. WEDDERBURN, Non-commutative domains of integrity, 07. reine u. angew. Math., I67 (1932), 129-I4I. Queen Mary College, University of London. Manuscrit refu le 15 juillet 1965.

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On the structure of theGL 2 of a ring

On the structure of theGL 2 of a ring

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On the structure of theGL 2 of a ring

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