Decompositions of reflexive modules

In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of lozenge (N1) which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is A congruent to A circle plus Z? The set theoretic hypothesis we made is the axiom lozenge (N1) which holds in particular in Godel's universe as shown by R. Jensen, see [8]. Here we want to prove a stronger result under the special continuum hypothesis (CH). The question in [8] relates 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 [8]. 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 [3] an R-module G is reflexive if the evaluation map sigma : 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 congruent to G circle plus R is natural, see [8, p. 455]. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [11]. Moreover we want to apply a model theoretic combinatorial theorem from [14] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to lozenge (N1) but holds under CH. This will simplify algebraic argument for G not congruent to G circle plus R.