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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nAn [[ideal|0MYSA6MYNA]] $I$ of $R$ is a *radical ideal* if $I = \\sqrt{I}$, where $\\sqrt{I}$ is the [[radical|437YRCNYRJ]] of $I$.\nEquivalently, for all $r \\in R$ and all integers $n \\ge 1$, if $r^n \\in I$ then $r \\in I$.\n\n## Remarks\n\nExample: $6\\Z$ is a radical ideal of $\\Z$; if $6$ divides $r^n$, then $2$ and $3$ divide $r$, so $6$ divides $r$.\nNon-example: $12\\Z$ is not radical, since $6^2 \\in 12\\Z$ and $6 \\notin 12\\Z$.\nEvery [[prime ideal|06HC6HK0Z8]] is radical: if $r^n \\in P$, primality applied $n - 1$ times gives $r \\in P$.\nThe ideal $I$ is radical if and only if the [[quotient ring|0VFW4Y69BT]] $R/I$ is [[reduced|SZHY975FFX]].\nThe radical $\\sqrt{J}$ of any ideal $J$ is a radical ideal.\n",
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