Free covers and minimal sets of generators (Q2078909)

Let \(R\) be a unital ring and \(M\) a right \(R\)-module. A free cover of \(M\) is an epimorphism \(\phi\colon F\to M\), where \(F\) is a free \(R\)-module, such that for every endomorphism \(f\) of \(F\), the equality \(\phi f=\phi\) implies that \(f\) is an automorphism. A set of generators for \(M\) is minimal if the submodule generated by any proper subset is properly contained in \(M\). In this paper, the author studies the relations between three classes of modules: \( \mathcal A\), modules which admit free covers; \(\mathcal B\), those in which every set of generators contains a minimal set of generators; and \(\mathcal C\), those in which every module has a minimal set of generators. The main results are that \(\mathcal A\bigcup\mathcal B\subseteq\mathcal C\); if \(R\) is local and right perfect, every \(R\)-module is in \(\mathcal A\bigcap\mathcal B\); if \(R\) is not right perfect, then \(\mathcal A\bigcap\mathcal B\) consists of the finitely generated modules \(M\) for which \(M/MJ\) a free \(R/J\)-module, where \(J\) is the Jacobson radical of \(R\).