Semi-perfect ring

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

definition

In the following, let R be a ring with 1 , J = J (R) the Jacobson radical .

A ring R is called semi-perfect if it has one of the following equivalent properties:

• Each simple R left / right module has a projective ceiling .
• Every finitely generated R left / right module has a projective ceiling.
• R / J is semi-simple, and any idempotent of R / J can be raised to R.
• There is a decomposition with pairwise orthogonal, local idempotents ${\displaystyle 1=\sum _{i=1}^{n}e_{i}}$ ${\displaystyle e_{1},\dots ,e_{n}}$

properties

• All leftartinian and rightartinian rings are semi-perfect.
• Each local ring is semi-perfect.
• A commutative ring R is semi-perfect if and only if R is a finite direct sum of local rings.
• If R is semi-perfect and I is an ideal of R, then the factor ring R / I is also semi-perfect.
• If R is semi-perfect and an idempotent, then eRe is also semi-perfect. ${\displaystyle e\in R}$