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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\n$R$ is *semiprime* if, for every $a \\in R$, the condition $aRa = 0$ implies $a = 0$.\n\n## Remarks\n\nEquivalent forms: $R$ has no nonzero [[nilpotent ideal|C2H26BYWHE]]; for every [[two-sided ideal|0DDA6CY7GJ]] $I$, $I^2 = 0$ implies $I = 0$; the [[prime radical|WF117EJ2HF]] of $R$ is $0$.\nExample: $\\Q \\times \\Q$ is semiprime: if $a = (a_1, a_2)$ and $aRa = 0$, then taking $r = 1$ gives $a_1^2 = a_2^2 = 0$, so $a = 0$.\nNon-example: in the ring $R$ of upper triangular $2 \\times 2$ matrices over $\\Q$, the matrix unit $a = e_{12}$ is nonzero and $e_{12} R e_{12} = 0$.\nEvery [[prime ring|Z9TJSZZYG1]] is semiprime.\nA commutative ring is semiprime if and only if it is [[reduced|SZHY975FFX]].\n",
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