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  "body": "\n## Summary\n\nA prime ring is a nonzero ring in which $aRb = 0$ forces $a = 0$ or $b = 0$.\nEquivalently, the product of two nonzero two-sided ideals is nonzero, that is, the zero ideal is a [[prime ideal|E8K2MN5V1E]].\nIt is the noncommutative replacement for an integral domain: a commutative ring is prime exactly when it is an integral domain.\nMatrix rings over prime rings are prime, so prime rings may have many zero divisors.\n\n## Explanations\n\n### L1 human\n\nIn an integral domain, $ab = 0$ forces $a = 0$ or $b = 0$.\nMatrix rings fail this, since two nonzero matrices can multiply to zero.\nA prime ring asks for something weaker: $a$ times anything times $b$ must not always be zero.\nThe ring $M_2(\\Q)$ is prime even though $e_{11} e_{22} = 0$, because $e_{11} e_{12} e_{22} = e_{12} \\ne 0$.\n",
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