Some nasty reflexive groups

by Goebel and Shelah. [GbSh:568]
Math Zeitschrift, 2001
In Almost Free Modules, Set-theoretic Methods, p. 455, Problem 12, Eklof and Mekler raised the question about the existence of dual abelian groups G which are not isomorphic to Z oplus G . Recall that G is a dual group if G cong 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 pi is such a homomorphism arising from a projection of the cartesian product, then D^* cong ker pi oplus Z . In all ``classical cases'' of groups D of infinite rank it turns out that D^* cong 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 ---> G^{**} is an isomorphism, hence G is the dual of G^* . Assuming the diamond axiom for aleph_1 we will construct a reflexive torsion-free abelian group of cardinality aleph_1 which is not isomorphic to Z oplus G . The result is formulated for modules over countable principal ideal domains which are not field.

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