Semiprimary ring

Summary

A semiprimary ring is a ring $R$ such that $R/J(R)$ is semisimple and the Jacobson radical $J(R)$ is nilpotent. Every left Artinian ring is semiprimary, because its Jacobson radical is nilpotent. Semiprimary rings are left and right perfect. They are the setting of the Hopkins–Levitzki theorem: a semiprimary ring is left Artinian exactly when it is left Noetherian.