Generalizations of Perfect, Semiperfect, and Semiregular Rings (original) (raw)

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 = Xsingular, we h a ve X = M . If there exists an epimorphism p : P ! M such that P is projective and Kerp i s -small in P , then we s a y 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 de ne R by R=SocRR = JacR=SocRR and show, among others, the following results: 1 R is the largest -small right ideal of R.