Semiperfect ring

Summary

A semiperfect ring is a semilocal ring in which idempotents lift modulo the Jacobson radical. Equivalently, every finitely generated left module has a projective cover. Equivalently again, $1$ is a sum of orthogonal idempotents $e_1, \ldots, e_n$ such that each corner ring $e_i R e_i$ is local. Left Artinian rings, local rings and perfect rings are semiperfect.