Decompositions of Reflexive Modules

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.