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).

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 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.
• 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]}

References

• Anderson, Frank W; Fuller, Kent R (1992), Rings and Categories of Modules, Springer, pp. 312–322, ISBN 0-387-97845-3  Cite uses deprecated parameters (help)
• 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, ISBN 0-387-95183-0, MR 1838439