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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThe *nilradical* of $R$ is the set\n$$N(R) = \\set{r \\in R : r^n = 0 \\text{ for some integer } n \\ge 1}$$\nof [[nilpotent elements|0N9R5NN6TW]] of $R$.\n\n## Remarks\n\n$N(R)$ is the [[radical|437YRCNYRJ]] $\\sqrt{(0)}$ of the zero ideal, so it is an ideal of $R$.\nExample: in $\\Z/12\\Z$, $N = \\set{0, 6}$; indeed $6^2 = 0$, and if $r^n \\equiv 0 \\pmod{12}$ then $2$ and $3$ divide $r$.\nNon-example: $2 \\notin N(\\Z/12\\Z)$, because $2^n$ is never divisible by $3$.\nThe quotient $R/N(R)$ has zero nilradical, that is, it is [[reduced|SZHY975FFX]].\n$N(R)$ is the intersection of all [[prime ideals|06HC6HK0Z8]] of $R$; see [[the nilradical as an intersection of primes|3CZ8PXR9JY]].\nOther notations are $\\operatorname{nil}(R)$ and $\\mathfrak{N}_R$.\n",
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