Semiperfect ring

A semi- perfect ring in the mathematical field of algebra is a ring over which every finitely generated left module has a projective cover. The term was introduced by Hyman Bass 1959/60.

Definition

The following is a ring with R 1, J = J ( R), the Jacobson radical.

A ring R is called semiperfect if he has one of the following equivalent properties:

• Each simple R-Links-/Rechtsmodul has a projective cover.
• Each R-Links-/Rechtsmodul finitely generated has a projective cover.
• R / J is semisimple, and every idempotent of R / J can be lifted to R.
• There exists a decomposition with pairwise orthogonal local idempotents

Properties

• Linksartinschen All and all rechtsartinschen rings are semiperfect.
• Each local ring is semiperfect.
• A commutative ring R if and only semi- perfect if R is a finite direct sum of local rings
• If R is semi- perfect, and I an ideal of R, then the factor ring R / I is semi- perfect.
• If R is semi- perfect and an idempotent, then eRe is semi- perfect.

• Ring ( algebra)
• Ring theory