2000/03/25 by Ruediger Goebel, Saharon Shelah, Goebel, Ruediger +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO
paper · pdf · doi:10.48550/arxiv.math/0003164
published as Math. Z. 237 No. 3 (2001) 547--559
arxiv created 2000/03/25 · arxiv updated 2009/11/30
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 ZD 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 aleph1 we construct a reflexive torsion-free abelian group of cardinality aleph1 which is not isomorphic to Z+G. The result is formulated for modules over countable principal ideal domains which are not field.