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Localization in Semifirs

Localization in Semifirs P. M. COHN 1. Introduction In [5] a method was described of forming the universal field of fractions for any semifir JR. In essence it consisted in taking the set 2 of all full matrices over R and forming the universal 2-inverting ring i? . This ring turns out to be the required field of fractions whenever R is a semifir. The process is reminiscent of the method of localization usual in commutative ring theory and it raises the question whether anything similar exists in the general case. A natural problem is the following: Given a semifir, if we invert certain elements, do we again get a semifir? An affirmative answer would lead to a simple proof that the group algebra kF of a free group F over a field A: is a semifir, as D. Eisenbud has pointed out. For the free algebra fc<X> is easily proved to be a semifir (even a fir) using the weak algorithm ([5], ch. 2), whereas the proof that kF is a semifir requires a more intricate argument involving free products [1, 2, 4]. But kF can be formed from k(X) (where F is the free group on X) by inverting the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/6.1.13
Publisher site
See Article on Publisher Site

Abstract

P. M. COHN 1. Introduction In [5] a method was described of forming the universal field of fractions for any semifir JR. In essence it consisted in taking the set 2 of all full matrices over R and forming the universal 2-inverting ring i? . This ring turns out to be the required field of fractions whenever R is a semifir. The process is reminiscent of the method of localization usual in commutative ring theory and it raises the question whether anything similar exists in the general case. A natural problem is the following: Given a semifir, if we invert certain elements, do we again get a semifir? An affirmative answer would lead to a simple proof that the group algebra kF of a free group F over a field A: is a semifir, as D. Eisenbud has pointed out. For the free algebra fc<X> is easily proved to be a semifir (even a fir) using the weak algorithm ([5], ch. 2), whereas the proof that kF is a semifir requires a more intricate argument involving free products [1, 2, 4]. But kF can be formed from k(X) (where F is the free group on X) by inverting the

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1974

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