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  "body": "\n## Summary\n\nA prime ideal of a noncommutative ring is a proper two-sided ideal $P$ with the following property.\nA product $AB$ of two-sided ideals lies in $P$ only when $A$ or $B$ does.\nFor commutative rings this is the usual [[prime ideal|00MHHZ9SNE]].\nIdeals are used instead of elements because, for $n \\ge 2$, the matrix ring $M_n(k)$ over a field has no proper ideal with the element-wise property.\nPrime ideals define the [[prime radical|WF117EJ2HF]] and are the kernels of maps onto [[prime rings|Z9TJSZZYG1]].\n\n## Explanations\n\n### L1 human\n\nFor integers, a prime $p$ divides $ab$ only if it divides $a$ or $b$.\nFor ideals of a noncommutative ring, this is asked for products of whole ideals, not of single elements.\nIn the ring $M_2(\\Q)$ the zero ideal is prime, because the only ideals are $0$ and the whole ring.\nBut it is not prime element-wise, since $e_{11} e_{22} = 0$ with $e_{11} \\ne 0$ and $e_{22} \\ne 0$.\n",
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