Generalizations of perfect, semiperfect, and semiregular rings (Q2711609)

For a ring \(R\) and a right \(R\)-module \(M\), a submodule \(N\subseteq M\) is said to be \(\sigma\)-small in \(M\) if, whenever \(N+X=M\) with \(M/X\) singular, then \(X=M\). If there exists an epimorphism \(p\colon P\to M\) such that \(P\) is projective and \(\text{Ker}(p)\) is \(\sigma\)-small in \(P\), then \(P\) is called a projective \(\sigma\)-cover of \(M\). This concept is a generalization of the concept of projective cover. From this the author defines the classes of \(\sigma\)-perfect, \(\sigma\)-semiperfect, and \(\sigma\)-semiregular rings. In the paper he obtains several characterizations of these rings.