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 ♦א1 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 ∼= A⊕Z? The set theoretic hypothesis we made is the axiom ♦א1 which holds in particular in Gödel’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 [1] an R-module G is reflexive if the evaluation map σ : 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 6∼= G ⊕ 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 ♦א1 but holds under CH. This will simplify algebraic argument for G 6∼= G⊕R. This work is supported by the project No. G-545-173.06/97 of the German-Israeli Foundation for Scientific Research & Development AMS subject classification: primary: 13C05, 13C10, 13C13, 20K15,20K25,20K30 secondary: 03E05, 03E35