Semilocal ring

Summary

A semilocal ring is a ring $R$ whose quotient $R/J(R)$ by the Jacobson radical is a semisimple ring. For a commutative ring this is the same as having finitely many maximal ideals, the commutative notion. Left Artinian rings and local rings are semilocal. Semilocal rings are the starting point for semiperfect, perfect and semiprimary rings, which add conditions on $J(R)$.