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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 ≥ ℵ 2 have non-free Whitehead modules even though they are not complete discrete valuation rings. A module M is a Whitehead module if Ext 1 R (M, R) = 0. The second author proved that the problem of whether every Whitehead Z-module… (More)

We investigate what Ext(A, Z) can be when A is torsion-free and Hom(A, Z) = 0. We thereby give an answer to a question of Golasi´nski and Gonçalves which asks for the divisible Abelian groups which can be the type of a co-Moore space.

It is proved consistent with ZFC + GCH that for every Whitehead group A of infinite rank, there is a Whitehead group H A such that Ext(H A , A) = 0. This is a strong generalization of the consistency of the existence of non-free Whitehead groups. A consequence is that it is undecidable in ZFC + GCH whether every Z-module has a ⊥ {Z}-precover. Moreover, for… (More)

We consider the question of which valuation domains (of cardinality ℵ 1) have non-standard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ZFC + GCH.

We study the classification of ω 1-separable groups using Ehrenfeucht-Fra¨ıssé games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming ♦.

- Paul C Eklof, L Aszl´o Fuchs
- 2007

We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns out to be equivalent to the question of when the tensor product of two Whitehead groups is Whitehead. We investigate what happens in different models of set theory.

Let R be a Dedekind domain. In [6], Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP + implies that C is not… (More)