Decompositions of reflexive modules

DOI10.1007/S000130050557zbMATH Open1014.03054arXivmath/0003165OpenAlexW2140765651MaRDI QIDQ5937384

Publication date: 30 October 2001

Published in: Archiv der Mathematik (Search for Journal in Brave)

Abstract: We continue [GbSh:568] (math.LO/0003164), proving a stronger result under the special continuum hypothesis (CH). The original question of Eklof and Mekler related to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H.Bass, an R-module G is reflexive if the evaluation map s:G-->G^{**} is an isomorphism. Here G^*=Hom(G,R) denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with G not cong G+R is natural. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [GbSh:568] (math.LO/0003164). Moreover we want to apply a model theoretic combinatorial theorem from [Sh:e] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to the diamond, but holds under CH.

Full work available at URL: https://arxiv.org/abs/math/0003165

zbMATH Keywords

dual module evaluation map reflexive module

Mathematics Subject Classification ID

Consistency and independence results (03E35) Continuum hypothesis and Martin's axiom (03E50) Applications of set theory (03E75) Torsion-free groups, infinite rank (20K20) Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups (20K30) Other special types of modules and ideals in commutative rings (13C13)