# Semiperfect ring

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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. - The category of finitely generated projective -modules is Krull-Schmidt.

## Examples

Examples of **semiperfect rings** include:

- Left (right) 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

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