Reflexive subgroups of the Baer-Specker group and Martin's axiom

Abstract

In two recent papers (math.LO/0003164 and math.LO/0003165) we answered a question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the set theoretical hypothesis of diamondsuit_{aleph_1} which holds in many models of set theory, respectively of the special continuum hypothesis (CH). The objects are reflexive modules over countable principal ideal domains R, which are not fields. 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 module of G. We proved the existence of reflexive R-modules G of infinite rank with G not cong G+R, which provide (even essentially indecomposable) counter examples to the question mentioned above. Is CH a necessary condition to find `nasty' reflexive modules? In the last part of this paper we will show (assuming the existence of supercompact cardinals) that large reflexive modules always have large summands. So at least being essentially indecomposable needs an additional set theoretic assumption. However the assumption need not be CH as shown in the first part of this paper. We will use Martin's axiom to find reflexive modules with the above decomposition which are submodules of the Baer-Specker module R^omega.