Semiprimitive ring

Summary

A semiprimitive ring is a ring whose Jacobson radical is zero. Equivalently, it is a subdirect product of primitive rings, or it has a faithful semisimple module. Every semiprimitive ring is semiprime, and a left Artinian ring is semiprimitive exactly when it is a semisimple ring. The ring $R/J(R)$ is always semiprimitive, which is why the Jacobson radical is the first thing one removes in the structure theory of rings.