# Perfect ring

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 Anderson and Fuller:
* 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

* Right or left Artinian rings, and semiprimary rings are known to be right-and-left perfect.
* The following is an example (due to Bass) of a local ring which is right but not left perfect. Let F be a field, and consider a certain ring of infinite matrices over F.
    Take the set of infinite matrices with entries indexed by $\mathbb {N} \times \mathbb {N}$, 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
       $R=\{f\cdot I+j\mid f\in F,j\in J\}\,$
    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.

### 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.
* An analogue of the Baer's criterion holds for projective modules.

## Semiperfect ring
### Definition

Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold:
* R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R.
* R has a complete orthogonal set e$_1$, ..., e$_n$ of idempotents with each e$_i$Re$_i$ a local ring.
* Every simple left (right) R-module has a projective cover.
* Every finitely generated left (right) R-module has a projective cover.
* The category of finitely generated projective $R$-modules is Krull-Schmidt.

### Examples

Examples of semiperfect rings include:
* Left (right) perfect rings.
* Local rings.
* Kaplansky's theorem on projective modules
* Left (right) Artinian rings.
* Finite dimensional k-algebras.

### 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

* ↑ Bass 1960.
* ↑ Anderson & Fuller 1992, p. 315.
* ↑ Lam 2001, pp. 345–346.