
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 .
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 :
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.