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

Abstract

In two recent papers (9, 10) we answered a question raised in the book by Eklof and Mekler (7, p. 455, Problem 12) under the set theoretical hypothesis of }@1 which holds in many models of set theory, respectively of the special continuum hypothesis (CH). The objects are re∞exive modules over countable principal ideal domains R, which are not flelds. Following H. Bass (1) an R-module G is re∞exive if the evaluation map ae : G i! G ⁄⁄ is an iso- morphism. Here G ⁄ = Hom(G;R) denotes the dual module of G. We proved the existence of re∞exive R-modules G of inflnite rank with G6 G'R, which provide (even essentially indecomposable) counter examples to the question (7, p. 455). Is CH a necessary condition to flnd 'nasty' re∞exive modules? In the last part of this paper we will show (assuming the existence of super- compact cardinals) that large re∞exive 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 flrst part of this paper. We will use Martin's axiom to flnd re∞exive modules with the above decomposition which are submodules of the Baer-Specker module R ! .