Construction of dual modules using Martin's Axiom

Several authors considered abelian groups, which can be represented as dual groups, see [P.C. Eklof, A.H. Mekler, Almost Free Modules—Set-Theoretic Methods, rev. ed., North-Holland Math. Library, North-Holland, 2002; R. Göbel, J. Trlifaj, Approximation Theory and Endomorphism Algebras, Walter de Gruyter, Berlin, 2005] for references. Recall that dual modules are those of the form G∗=HomR(G,R). We will work in the category of R-modules, over countable PIDs with a multiplicatively closed subset S such that R is Hausdorff in its natural S-topology. Using Martin's Axiom (MA), we represent a large class of modules which are submodules of P=Rω as dual modules. Martin's Axiom is mainly used to reduce the problem of solving infinite systems of linear equations over R to the finite one (see Step Lemma 4.10). An analysis in Section 4 will show that being a dual module of a submodule H⊆P requires two necessary conditions (Definition 4.7); we will say that H is admissible in this case. Conversely we will show (Theorem 4.13) that under (MA) and negation of (CH) precisely these modules (of size <ℵ02) are dual modules of pure modules G sandwiched between R(ω) and its S-adic closure in P. In the last section of the paper we strengthen this result by the additional demand that the endomorphism ring of G is ‘minimal,’ hence G becomes also essentially rigid.