Some Nasty reflexive groups

Abstract.

In {\it 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 ${\mathbb Z} \oplus G$. Recall that G is a dual group if $G \cong D^*$ for some group D with $D^* ={\rm Hom}(D,{\mathbb Z})$. The existence of such groups is not obvious because dual groups are subgroups of cartesian products ${\mathbb Z}^{D}$ and therefore have very many homomorphisms into ${\mathbb Z}$. If $\pi$ is such a homomorphism arising from a projection of the cartesian product, then $D^*\cong{\rm ker} \pi \oplus{\mathbb Z}$. In all `classical cases' of groups {\it D} of infinite rank it turns out that $D^* \cong{\rm ker} \pi$. Is this always the case? Also note that reflexive groups G in the sense of H. Bass are dual groups because by definition the evaluation map $\sigma: G \longrightarrow G^{**}$ is an isomorphism, hence G is the dual of $G^*$. Assuming the diamond axiom for $\aleph_1 (\Diam_{\aleph_1})$ we will construct a reflexive torsion-free abelian group of cardinality $\aleph_1$ which is not isomorphic to ${\mathbb Z} \oplus G$. The result is formulated for modules over countable principal ideal domains which are not field.