# perfect and semiperfect rings

A ring $R$ is called left/right perfect if for any left/right $R$-module $M$ there exists a projective cover $p:P\to M$.

A ring $R$ is called left/right semiperfect if for any left/right finitely-generated $R$-module $M$ there exists a projective cover $p:P\to M$.

It can be shown that there are rings which are left perfect, but not right perfect. However being semiperfect is left-right symmetric property.

Some examples of semiperfect rings include:

1. 1.

perfect rings;

2. 2.

left/right Artinian rings;

3. 3.

finite-dimensional algebras over a field $k$.

Title perfect and semiperfect rings PerfectAndSemiperfectRings 2013-03-22 19:17:56 2013-03-22 19:17:56 joking (16130) joking (16130) 4 joking (16130) Definition msc 16D40