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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\n$R$ is a *prime ring* if $R \\ne 0$ and, for all $a, b \\in R$, the condition $aRb = 0$ implies $a = 0$ or $b = 0$.\nHere $aRb = \\set{arb : r \\in R}$.\n\n## Remarks\n\nExample: $M_n(k)$ over a field $k$ is prime.\nIf $a \\ne 0$ 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 $a_{ij} b_{kl} \\ne 0$.\nNon-example: $k \\times k$ is not prime, since $(1, 0)\\, r\\, (0, 1) = 0$ for every $r \\in k \\times k$.\nEquivalent forms: the zero ideal is a [[prime ideal|E8K2MN5V1E]] of $R$; or, for two-sided ideals $A, B$ of $R$, $AB = 0$ implies $A = 0$ or $B = 0$.\nA commutative ring is prime if and only if it is an integral domain.\nEvery [[primitive ring|8F6EHDVJ62]] is prime.\n",
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