Construction of dual modules using Martin's axiom.
It is a challenging open problem to classify all Abelian groups \(G\) that are dual groups, which means \(G=H^*=\Hom(H,\mathbb{Z})\) for some Abelian group \(H\). The authors attack this problem using additional set-theory by applying Martin's axiom. Working in the category of \(R\)-modules over countable principal ideal domains \(R\) the authors are able to represent large classes of modules that are submodules of \(R^\omega\) as dual modules. The idea is to reduce the problem of solving infinite systems of linear equations over \(R\) to finite ones using Martin's axiom. The obtained modules are sandwiched between \(R^{(\omega)}\) and its closure inside \(R^\omega\). Going one step further the authors are also able to demand even more on the groups by requiring that the endomorphism ring is minimal, i.e. \(\text{End}(G)=R\oplus\text{Fin}(G)\). The paper is well-written and can certainly be recommended to anyone who is interested in dual groups and the use of Martin's axiom.
scientific article; zbMATH DE number 5365171