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 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.
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 e₁, ..., 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 $R$ -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, ISSN 0002-9947, JSTOR 1993568, MR 0157984, doi:10.2307/1993568
Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439

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