# Perfect ring

In the area of abstract algebra known as ring theory, a **left perfect ring** is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in (Bass 1960).

A **semiperfect ring** is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.

The following equivalent definitions of a left perfect ring *R* are found in (Anderson,Fuller & 1992, p.315):

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

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.