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  "body": "\n## Summary\n\nA radical ideal of a commutative ring is an [[ideal|0MYSA6MYNA]] that equals its own [[radical|437YRCNYRJ]].\nEquivalently, if some power of an element lies in the ideal, then the element itself lies in it.\nEvery [[prime ideal|06HC6HK0Z8]] is radical.\nBy [[Hilbert's Nullstellensatz|D85VZWS9QK]], the radical ideals of a polynomial ring over an algebraically closed field correspond to the algebraic sets.\n\n## Explanations\n\n### L1 human\n\nA radical ideal has no \"repeated factors\".\nIn $\\Z$, the ideal $n\\Z$ with $n \\ge 1$ is radical exactly when $n$ is square-free, such as $6$ or $30$.\nThe ideal $12\\Z$ is not radical: $6^2 = 36$ lies in it, but $6$ does not.\nThe quotient by a radical ideal is a [[reduced ring|SZHY975FFX]], a ring without nonzero nilpotent elements.\n",
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