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  "body": "\n## Summary\n\nA semiprime ring is a ring in which $aRa = 0$ forces $a = 0$.\nEquivalently, it has no nonzero nilpotent two-sided ideals, or its [[prime radical|WF117EJ2HF]] is zero.\nSemiprime rings are the subdirect products of [[prime rings|Z9TJSZZYG1]], and a commutative ring is semiprime exactly when it is [[reduced|SZHY975FFX]].\nThey form the natural setting of Goldie's theorem and of much of the theory of polynomial identity rings.\n\n## Explanations\n\n### L1 human\n\nA reduced commutative ring has no nonzero nilpotent elements.\nIn noncommutative rings this is too strong, since matrix rings have nilpotent matrices.\nA semiprime ring instead has no nonzero ideal whose elements all multiply to zero in long enough products.\nFor example $M_2(\\Q)$ is semiprime, while the ring of upper triangular $2 \\times 2$ matrices is not, because its strictly upper triangular matrices form an ideal with square zero.\n",
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