Semiperfect ring

In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric.

Contents

1 Definition
2 Examples
3 Properties
4 References

Definition

Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold:

R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R.
R has a complete orthogonal set e₁, ..., e_n of idempotents with each e_i R e_i a local ring.
Every simple left (right) R-module has a projective cover.
Every finitely generated left (right) R-module has a projective cover.
The category of finitely generated projective $R$ -modules is Krull-Schmidt.

Examples

Examples of semiperfect rings include:

Properties

Since a ring R is semiperfect iff every simple left R-module has a projective cover, every ring Morita equivalent to a semiperfect ring is also semiperfect.

References

Anderson, Frank Wylie; Fuller, Kent R (1992). Rings and Categories of Modules. Springer. ISBN 0387978453. http://books.google.com/books?id=PswhrD_wUIkC. Retrieved 2007-03-27.