Perfect ring

Summary

A left perfect ring is a ring $R$ such that $R/J(R)$ is semisimple and the Jacobson radical $J(R)$ is left T-nilpotent. By Bass's theorem P, these are exactly the rings over which every left module has a projective cover. Left perfect rings are semiperfect, and semiprimary rings, in particular left Artinian rings, are left and right perfect. Left perfect and right perfect are different conditions.