# 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.

## Perfect ring

### 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

- 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 ℕ× ℕ, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by . 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)

### 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}*R e*_{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 -modules is Krull-Schmidt.

### Examples

Examples of semiperfect rings include:

- Left (right) perfect rings.
- Local rings.
- 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.

## References

- Anderson, Frank W; Fuller; Kent R (1992),
*Rings and Categories of Modules*, Springer, pp. 312–322, ISBN 0-387-97845-3 - Bass, Hyman (1960), "Finitistic dimension and a homological generalization of semi-primary rings",
*Transactions of the American Mathematical Society*,**95**(3): 466–488, doi:10.2307/1993568, ISSN 0002-9947, JSTOR 1993568, MR 0157984 - Lam, T. Y. (2001),
*A first course in noncommutative rings*, Graduate Texts in Mathematics,**131**(2 ed.), New York: Springer-Verlag, pp. xx+385, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439