Generalizations of perfect, semiperfect, and semiregular rings. (English) Zbl 0994.16016

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.


16L30 Noncommutative local and semilocal rings, perfect rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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