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Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an ℵ 1-free R-module G of rank ℵ 1 with endomorphism algebra End R G = A. Clearly the result does not hold for fields. Recall that an R-module is ℵ 1-free if all its countable submodules are free, a condition… (More)

In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of ♦ ℵ 1 which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is A ∼ = A ⊕ Z? The set theoretic hypothesis we made is… (More)

In Almost Free Modules, Set-theoretic Methods, Eklof and Mekler [5, p. 455, Problem 12] raised the question about the existence of dual abelian groups G which are not isomorphic to Z ⊕ G. Recall that G is a dual group if G ∼ = D * for some group D with D * = Hom (D, Z). The existence of such groups is not obvious because dual groups are subgroups of… (More)

In the first part of this paper we introduce a simplified version of a new Black Box from Shelah [11] which can be used to construct complicated ℵn-free abelian groups for any natural number n ∈ N. In the second part we apply this prediction principle to derive for many commutative rings R the existence of ℵn-free R-modules M with trivial dual M * = 0,… (More)