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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $R$.\nThe *radical* of $I$ is the set\n$$\\sqrt{I} = \\set{r \\in R : r^n \\in I \\text{ for some integer } n \\ge 1}.$$\n\n## Remarks\n\nThe set $\\sqrt{I}$ is an ideal of $R$ that contains $I$: if $a^m \\in I$ and $b^n \\in I$, then every term of the binomial expansion of $(a + b)^{m + n - 1}$ lies in $I$.\nExample: in $\\Z$, $\\sqrt{12\\Z} = 6\\Z$, since $6^2 = 36 \\in 12\\Z$, and if $r^n \\in 12\\Z$ then the primes $2$ and $3$ divide $r$.\nNon-example: $4 \\notin \\sqrt{12\\Z}$, because no power of $4$ is divisible by $3$.\nThe radical of the zero ideal is the [[nilradical|QQDN6EYQNJ]] of $R$.\nAn ideal equal to its radical is a [[radical ideal|S4GQN5V54X]].\nOther notations are $\\operatorname{rad}(I)$ and $r(I)$.\n",
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