# Generalizations of Perfect, Semiperfect, and Semiregular Rings

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 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(R R ) = Jac(R/Soc(R R)) 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(R R ) is a right perfect ring and idempotents of R/δ(R) lift to idempotents of R. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

# Generalizations of Perfect, Semiperfect, and Semiregular Rings

, Volume 7 (3) – Jan 1, 2000
14 pages

/lp/springer-journals/generalizations-of-perfect-semiperfect-and-semiregular-rings-xbdNb3uiV9
Publisher
Springer Journals
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s10011-000-0305-9
Publisher site
See Article on Publisher Site

### 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(R R ) = Jac(R/Soc(R R)) 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(R R ) is a right perfect ring and idempotents of R/δ(R) lift to idempotents of R.

### Journal

Algebra ColloquiumSpringer Journals

Published: Jan 1, 2000