Prime ring

Remarks

Example: $M_n(k)$ over a field $k$ is prime. If $a \ne 0$ and $b \ne 0$, choose entries $a_{ij} \ne 0$ and $b_{kl} \ne 0$; then $e_{jk} \in R$ and the $(i, l)$ entry of $a e_{jk} b$ is $a_{ij} b_{kl} \ne 0$. Non-example: $k \times k$ is not prime, since $(1, 0)\, r\, (0, 1) = 0$ for every $r \in k \times k$. Equivalent forms: the zero ideal is a prime ideal of $R$; or, for two-sided ideals $A, B$ of $R$, $AB = 0$ implies $A = 0$ or $B = 0$. A commutative ring is prime if and only if it is an integral domain. Every primitive ring is prime.