Prime ring

Summary

A prime ring is a nonzero ring in which $aRb = 0$ forces $a = 0$ or $b = 0$. Equivalently, the product of two nonzero two-sided ideals is nonzero, that is, the zero ideal is a prime ideal. It is the noncommutative replacement for an integral domain: a commutative ring is prime exactly when it is an integral domain. Matrix rings over prime rings are prime, so prime rings may have many zero divisors.