Some Nasty reflexive groups

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 . Recall that G is a dual group if for some group D with . The existence of such groups is not obvious because dual groups are subgroups of cartesian products and therefore have very many homomorphisms into . If is such a homomorphism arising from a projection of the cartesian product, then . In all `classical cases' of groups {\it D} of infinite rank it turns out that . 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 is an isomorphism, hence G is the dual of . Assuming the diamond axiom for we will construct a reflexive torsion-free abelian group of cardinality which is not isomorphic to . The result is formulated for modules over countable principal ideal domains which are not field. Received July 1, 1999; in final form January 26, 2000 / Published online April 12, 2001