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 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):

For a left perfect ring R:

Let R be ring. 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.

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