Perfect ring

Perfect ring

Definitions

• Every left R module has a projective cover.
• R/J is semisimple and J is left T-nilpotent, where J is the Jacobson radical of R.
• R satisfies the descending chain condition on principal right ideals.
• Every flat left R-module is projective.
• R/J 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 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.

  Properties

• 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

• R/J is semisimple and idempotents lift modulo J, where J 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 R-module has a projective cover.
• Every finitely generated left R-module has a projective cover.
• The category of finitely generated projective -modules is Krull-Schmidt.

  Examples

• Left perfect rings.
• Local rings.
• Kaplansky's theorem on projective modules
• Left Artinian rings.
• Finite dimensional k-algebras.

  Properties