Some Nasty Reflexive Groups

Abstract

In "Almost Free Modules, Set-theoretic Methods", Eklof and Mekler 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 cartesian products Z^D and therefore have very many homomorphisms into Z. If p is such a homomorphism arising from a projection of the cartesian product, then D^* ~ ker(p)+Z. In all `classical cases' of groups D of infinite rank it turns out that D^* ~ ker(p). 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 s:G-->G^{**} is an isomorphism, hence G is the dual of G^*. Assuming the diamond axiom for aleph_1 we construct a reflexive torsion-free abelian group of cardinality aleph_1 which is not isomorphic to Z+G. The result is formulated for modules over countable principal ideal domains which are not field.