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S. Shelah (1975)
A compactness theorem for singular cardinals, free algebras, Whitehead problem and tranversalsIsrael Journal of Mathematics, 21
Otto Gerstner, Ludger Kaup, H. Weidner (1969)
Whitehead-Moduln abzählbaren Ranges über HauptidealringenArchiv der Mathematik, 20
revision:2000-11-24 modified:2000-11-27 WHITEHEAD MODULES OVER LARGE PRINCIPAL IDEAL DOMAINS 5
S. Shelah (1974)
Infinite abelian groups, whitehead problem and some constructionsIsrael Journal of Mathematics, 18
Define ψ(w δ,n ) to be t ωδ+σ δ +n where ωδ + σ δ is larger than any µ which occurs in any p δ,k or s δ,k, for k ∈ ω, < d. Define T α as before and choose δ ∈ S such that T δ ∩ ω 1 ⊆ ωδ
(1969)
Arch. Math. (Basel)
Paul Eklof (1990)
Almost free modules
Ruediger Goebel, S. Shelah (1999)
COTORSION THEORIES AND SPLITTERSTransactions of the American Mathematical Society, 352
P. Eklof (1998)
Non-perfect rings and a theorem of Eklof and Shelah
L. Fuchs, S. Shelah, T. Becker (1989)
Whitehead Modules over Domains, 1
Shelah) Institute of Mathematics
S. Shelah (1980)
Whitehead groups may not be free even assuming ch, IIIsrael Journal of Mathematics, 35
Abstract. We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.'s of size d2 have non-free Whitehead modules even though they are not complete discrete valuation rings. 2000 Mathematics Subject Classi®cation: 03E75, 13D07. 1 A module M is a Whitehead module if ExtR MY R 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC GCH (cf. [5], [6], [7]). This was extended in [1] to modules over principal ideal domains of cardinality at most d1 . Here we consider the Whitehead problem for modules over principal ideal domains (p.i.d.'s) of cardinality b d1 . If R is any p.i.d. which is not a complete discrete valuation ring, then an R-module of countable rank is Whitehead if and only if it is free (cf. [3]). On the other hand, if R is a complete discrete valuation ring, then it is cotorsion and hence every torsion-free R-module is a Whitehead module (cf. [2, XII.1.17]). It will be convenient to decree that a ®eld is not a p.i.d. and to use the term ``slender'' to designate a p.i.d. which is not
Forum Mathematicum – de Gruyter
Published: Apr 15, 2002
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