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  "body": "\n## Summary\n\nThe radical of an [[ideal|0MYSA6MYNA]] $I$ of a commutative ring is the set of elements some power of which lies in $I$.\nIt is again an ideal, and it contains $I$.\nThe radical is the intersection of the [[prime ideals|06HC6HK0Z8]] that contain $I$.\nIt is central in the link between ideals and geometry given by [[Hilbert's Nullstellensatz|D85VZWS9QK]].\n\n## Explanations\n\n### L1 human\n\nThe radical forgets multiplicities.\nIn $\\Z$, the ideal $12\\Z$ records the number $12 = 2^2 \\cdot 3$, and its radical is $6\\Z$, which only records the primes $2$ and $3$.\nAn integer $r$ has a power divisible by $12$ exactly when $r$ is divisible by $2$ and by $3$.\nFor polynomials, the ideals $(x^2)$ and $(x)$ have the same zero set, and the radical of $(x^2)$ is $(x)$.\n",
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