Generalizations of Perfect, Semiperfect, and Semiregular Rings

Abstract

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P → M such that P is projective and Ker(p) is δ-small in P, then we say that P is a projective δ-cover of M. A ring R is called δ-perfect (resp., δ-semiperfect, δ-semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective δ-cover. The class of all δ-perfect (resp., δ-semiperfect, δ-semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of δ-perfect, δ-semiperfect, and δ-semiregular rings. We define δ(R) by δ(R)/Soc(RR) = Jac(R/Soc(RR)) and show, among others, the following results: (1) δ(R) is the largest δ-small right ideal of R. (2) R is δ-semiregular if and only if R/δ(R) is a von Neumann regular ring and idempotents of Rδ(R) lift to idempotents of R. (3) R is δ-semiperfect if and only if R/δ(R) is a semisimple ring and idempotents of R/δ(R) lift to idempotents of R. (4) R is δ-perfect if and only if R/Soc(RR) is a right perfect ring and idempotents of R/δ(R) lift to idempotents of R.