Semiprime ring

Summary

A semiprime ring is a ring in which $aRa = 0$ forces $a = 0$. Equivalently, it has no nonzero nilpotent two-sided ideals, or its prime radical is zero. Semiprime rings are the subdirect products of prime rings, and a commutative ring is semiprime exactly when it is reduced. They form the natural setting of Goldie's theorem and of much of the theory of polynomial identity rings.