Some nasty reflexive groups

DOI10.1007/PL00004879zbMATH Open0988.20044arXivmath/0003164OpenAlexW2146511592MaRDI QIDQ5947799

Publication date: 22 October 2001

Published in: Mathematische Zeitschrift (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/math/0003164

zbMATH Keywords

reflexive modules principal ideal domains diamond axiom dual Abelian groups

Mathematics Subject Classification ID

Projective and free modules and ideals in commutative rings (13C10) Consistency and independence results (03E35) Structure, classification theorems for modules and ideals in commutative rings (13C05) Torsion-free groups, infinite rank (20K20) Other combinatorial set theory (03E05) Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups (20K30) Direct sums, direct products, etc. for abelian groups (20K25) Other special types of modules and ideals in commutative rings (13C13)