Perfect ring

Definitions

The following equivalent definitions of a left perfect ring R are found in Anderson 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.

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 ${\displaystyle \mathbb {N} \times \mathbb {N} }$, 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]

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. ^{[citation needed]}

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_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

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.

Basic ring

For a fixed semiperfect ring R, there is (up to a ring isomorphism) a canonical representative of the class of rings Morita equivalent to R called a basic ring for R. A semiperfect ring is basic if and only if R/J(R) is a direct product of division rings.

Given a complete orthogonal set e₁, ..., e_n of local idempotents, we can assume ordering of this set in such a way that the for some i≤n the right modules e₁R, ..., e_iR form a complete set of finitely generated projective right modules. The idempotent e=e₁+ ... + e_i is then called a basic idempotent and the corner ring B:=eRe isomorphic to End_R(e₁R + ...+ e_i R) is the basic ring for R.

The map I→IR defines embedding of the lattice of right ideals of B into that of R. The map I→RIR defines an isomorphism of the lattice of ideals of B and the lattice of ideals of R. This isomorphism respects multiplication of ideals. ^[4]