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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $I$ be a [[left ideal|5N2CJZ47D4]], right ideal or [[two-sided ideal|0DDA6CY7GJ]] of $R$.\nFor $n \\ge 1$, let $I^n$ be the additive subgroup of $R$ generated by all products $a_1 a_2 \\cdots a_n$ with $a_1, \\dots, a_n \\in I$.\nThe ideal $I$ is *nilpotent* if $I^n = 0$ for some $n \\ge 1$.\n\n## Remarks\n\nEvery nilpotent ideal is a [[nil ideal|NZ1G9WRWFB]], since $I^n = 0$ gives $a^n = 0$ for all $a \\in I$.\nExample: let $R$ be the ring of upper triangular $2 \\times 2$ matrices over a field $K$ and let $I$ be the set of its matrices with zero diagonal.\n$I$ is a two-sided ideal, and the product of two matrices of the form $\\begin{pmatrix} 0 \u0026 x \\\\ 0 \u0026 0 \\end{pmatrix}$ is zero, so $I^2 = 0$.\nNon-example: let $R = \\Z[x_1, x_2, \\dots] / (x_1, x_2^2, x_3^3, \\dots)$ and let $I$ be the ideal generated by the images of all $x_k$.\n$I$ is nil, since it is generated by nilpotent elements of a commutative ring.\nIt is not nilpotent, because for every $n$ the product $x_{n+1}^{n}$ of $n$ elements of $I$ is nonzero in $R$.\n",
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  "name": "Nilpotent ideal",
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