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  "body": "\n## Summary\n\nA nilpotent ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal $I$ such that some power $I^n$ is zero.\nThis means that every product of $n$ elements of $I$ vanishes, a uniform version of being a [[nil ideal|NZ1G9WRWFB]].\nEvery nilpotent ideal is nil, but a nil ideal need not be nilpotent.\nA ring is called semiprime when it has no nonzero nilpotent ideals.\nThe Jacobson radical of a left Artinian ring is a nilpotent ideal.\n\n## Explanations\n\n### L1 human\n\nIn a nil ideal each element dies after enough self-multiplications.\nIn a nilpotent ideal a single number $n$ works for all of them at once: any product of $n$ elements of the ideal, equal or not, is zero.\nIn the ring of upper triangular $2 \\times 2$ matrices, the matrices with zero diagonal form a nilpotent ideal.\nThe product of any two of them is zero.\n",
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