Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -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 -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

(1)	(R) is the largest -small right ideal of R.
(2)	R is -semiregular if and only if R/(R) is a von Neumann regular ring and idempotents of R(R) lift to idempotents of R.
(3)	R is -semiperfect if and only if R/(R) is a semisimple ring and idempotents of R/(R) lift to idempotents of R.
(4)	R is -perfect if and only if R/Soc(RR) is a right perfect ring and idempotents of R/(R) lift to idempotents of R.

The research was partially supported by the NSERC of Canada under Grant OGP0194196.2000 Mathematics Subject Classification: 16L30, 16E50