Perfect ring

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (March 2016) (Learn how and when to remove this message)

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.^[1]

A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.

The following equivalent definitions of a left perfect ring R are found in Aderson and Fuller:^[2]

• 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.

• 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 ${\displaystyle \mathbb {N} \times \mathbb {N} }$, and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by ${\displaystyle J}$. Also take the matrix ${\displaystyle I\,}$ with all 1's on the diagonal, and form the set

${\displaystyle R=\{f\cdot I+j\mid f\in F,j\in J\}\,}$

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.^[3]

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. ^{[citation needed]}

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_iRe_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 ${\displaystyle R}$-modules is Krull-Schmidt.

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.

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.