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: