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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nLet $I$ be a [[left ideal|5N2CJZ47D4]], right ideal or [[two-sided ideal|0DDA6CY7GJ]] of $R$.\nThe ideal $I$ is *nil* if every element of $I$ is [[nilpotent|0N9R5NN6TW]].\nThis means that for each $a \\in I$ there is an integer $n \\ge 1$ with $a^n = 0$.\n\n## Remarks\n\nExample: in $\\Z/8\\Z$, the ideal $I = 2\\Z/8\\Z$ is nil.\nThe check: every element of $I$ is $2b$ for some residue $b$, and $(2b)^3 = 8b^3 = 0$.\nNon-example: in $\\Z$, the ideal $2\\Z$ is not nil, since $2^n \\ne 0$ for all $n \\ge 1$.\nIn a commutative ring the nil ideals are the ideals contained in the [[nilradical|2K5GKVZR78]].\nThe exponent $n$ may depend on the element, and there need not be a common bound.\n",
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  "markdown": "https://tamnd.github.io/math2/node/WKG2DZDEBT.md",
  "msc": [
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  "name": "Nil ideal",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/WKG/WKG2DZDEBT.md",
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