Perfect ring

This article is about perfect rings as introduced by Hyman Bass. For perfect rings of characteristic p generalizing perfect fields, see perfect field.

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.

Perfect ring


The following equivalent definitions of a left perfect ring R are found in (Anderson,Fuller 1992, p.315):


Take the set of infinite matrices with entries indexed by ℕ× ℕ, and which only have finitely many nonzero entries above the diagonal, and denote this set by J. Also take the matrix with all 1's on the diagonal, and form the set
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. (Lam 2001, p.345-346)


For a left perfect ring R:

Semiperfect ring


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.


This article is issued from Wikipedia - version of the 9/24/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.