Prime ring

A prime ring is a ring $R$ such that for any nonzero two-sided ideals $A,B\subseteq R$, one has $AB\neq 0$, where $AB$ denotes the product of ideals . Equivalently: if $A$ and $B$ are ideals with $AB=0$, then $A=0$ or $B=0$.

In the commutative case, primeness is closely related to being an integral domain : if $R$ is commutative (and $1\neq 0$), the “ideal” condition forces $ab=0$ to imply $a=0$ or $b=0$.

Examples:

• If $D$ is a division ring and $n\ge 1$, then $M_n(D)$ is a prime ring.
• Any integral domain is a prime ring (commutative case).
• A direct product $R_1\times R_2$ with $R_1,R_2\neq 0$ is not prime: $(R_1\times 0)(0\times R_2)=0$.