Semiperfect ring

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.

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 "e"1, ..., "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.

Examples

Examples of semiperfect rings include:

* Perfect rings.
* Local rings.
* Left (right) Artinian rings.
* Finite dimensional "k"-algebras.

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

*cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|date = 1992|isbn = 0387978453|url = http://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27


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