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. 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:

Examples

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.

Citations