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's book. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric. Perfect ring. Definitions. The following equivalent definitions of a left  perfect ring "R" are found in Aderson and Fuller: Properties. For a left perfect ring "R": Semiperfect ring. Definition. Let "R" be ring. Then "R" is semiperfect if any of the following equivalent conditions hold: Examples. Examples of semiperfect rings include: 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.