# Perfect ring

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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).

## Definitions

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

• Every left R module has a projective cover.
• R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R.
• (Bass' Theorem P) R satisfies the descending chain condition on principal right ideals. (There is no mistake, this condition on right principal ideals is equivalent to the ring being left perfect.)
• Every flat left R-module is projective.
• R/J(R) is semisimple and every non-zero left R module contains a maximal submodule.
• R contains no infinite orthogonal set of idempotents, and every non-zero right R module contains a minimal submodule.

## Examples

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 $I\,$ with all 1's on the diagonal, and form the set
$R=\{f\cdot I+j\mid f\in F, j\in J \}\,$
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)

## Properties

For a left perfect ring R:

• From the equivalences above, every left R module has a maximal submodule and a projective cover, and the flat left R modules coincide with the projective left modules.
• R is a semiperfect ring, since one of the characterizations of semiperfect rings is: "All finitely generated left R modules have projective covers."
• An analogue of the Baer's criterion holds for projective modules.[citation needed]