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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nAn element $x \\in R$ is nilpotent if there is a positive integer $n$ such that $x^n = 0$.\n\n## Remarks\n\nIn $\\mathbb{Z}/8\\mathbb{Z}$, the class of $2$ is nilpotent, because $2^3 = 8 = 0$.\nThe matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ is nilpotent: its square is the zero matrix.\nIn a nonzero ring, a nilpotent element is a [[zero divisor|0K26YKNBSA]]: if $n$ is the least integer with $x^n = 0$, then $x \\cdot x^{n-1} = 0$ and $x^{n-1} \\neq 0$.\nAn [[integral domain|09S3PFBEPM]] has no nilpotent element except $0$.\n\n## Explanations\n\n### L1 human\n\nA nilpotent element is not zero, but some power of it is zero.\nNumbers do not behave like this.\nIn the integers modulo $8$, the element $2$ does: $2 \\cdot 2 \\cdot 2 = 8$, and $8$ is $0$ modulo $8$.\n",
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