Some nasty reflexive groups (Q5947799)

In the book ``Almost free modules. Set-theoretic methods'' of \textit{P. Eklof} and \textit{A. H. Mekler} [North-Holland, Amsterdam (1990; Zbl 0718.20027)], Problem 12 raises the question concerning the existence of a dual Abelian group \(G\) (i.e. a group with \(G\cong\Hom(\mathbb{Z},D)\) for some group \(D\)) which is not isomorphic to the direct sum \(\mathbb{Z}\oplus G\). The purpose of the paper is to give an affirmative answer to this problem. More precisely, assuming the diamond axiom for \(\aleph_1\) the main Theorem states, that if \(R\) is a countable principal ideal domain which is not a field, then there exists a reflexive \(R\)-module \(G\) of cardinality \(\aleph_1\) such that \(G\ncong R\oplus G\).