Some Nasty Reflexive Groups

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 groupD withD = Hom (D,Z). The existence of such groups is not obvious because dual groups are subgroups of cartesian products Z and therefore have very many homomorphisms into Z. If π is such a homomorphism arising from a projection of the cartesian product, then D ∼= kerπ ⊕ Z. In all ‘classical cases’ of groups D of infinite rank it turns out that D ∼= ker π. 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 σ : G −→ G is an isomorphism, hence G is the dual of G. Assuming the diamond axiom for א1 (♦א1) we will construct a reflexive torsion-free abelian group of cardinality א1 which is not isomorphic to Z ⊕ G. The result is formulated for modules over countable principal ideal domains which are not field.