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 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 π is such a homomorphism arising from a projection of the cartesian product, then D* ≅ ker π ⊕ 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.