Perfect ring explained

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

Take the set of infinite matrices with entries indexed by

N x N

, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by

J

. Also take the matrix

I

with all 1's on the diagonal, and form the set

R=\{fI+j\midf\inF,j\inJ\}

It can be shown that R is a ring with identity, whose Jacobson radical is J. Furthermore R/J is a field, so that R is local, and R is right but not left perfect.

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:

R

-modules is Krull-Schmidt.

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.