In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over 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.
[1] A semiperfect ring is a ring over which every finitely generated left module has a projective cover.
Then R is semiperfect if any of the following equivalent conditions hold: Examples of semiperfect rings include: 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.