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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThen the [[nilradical|QQDN6EYQNJ]] of $R$ is the intersection of all [[prime ideals|06HC6HK0Z8]] of $R$:\n$$N(R) = \\bigcap_{P \\text{ prime}} P.$$\n\n## Remarks\n\nOne inclusion is direct: if $r^n = 0$, then $r^n$ lies in every prime ideal $P$, so $r \\in P$.\nFor the other inclusion, let $f$ be an element that is not [[nilpotent|0N9R5NN6TW]].\nThen the set $S = \\set{1, f, f^2, \\ldots}$ does not contain $0$, and [[Zorn's lemma|04RXTQA5ZA]] gives an ideal $P$ that is maximal among the ideals disjoint from $S$.\nSuch an ideal is prime, and $f \\notin P$.\nExample: in $\\Z/12\\Z$ the prime ideals are $2\\Z/12\\Z$ and $3\\Z/12\\Z$, and their intersection $6\\Z/12\\Z = \\set{0, 6}$ is the set of nilpotent elements.\nIf $R$ is the zero ring, it has no prime ideals, the empty intersection is $R$, and every element of $R$ is nilpotent, so the statement still holds.\nApplied to $R/I$, the statement shows that the [[radical|437YRCNYRJ]] of an ideal $I$ is the intersection of the prime ideals that contain $I$.\n\n## Explanations\n\n### L1 human\n\nAn element of a commutative ring is nilpotent exactly when it lies in every prime ideal.\nBeing nilpotent is an algebraic property: some power is zero.\nLying in every prime ideal is a geometric property: the element, seen as a function on the [[spectrum|FB45BHHA2J]], vanishes at every point.\nThe theorem says that these two properties agree.\n\n### L2 human\n\nThe theorem is the starting point of the link between ideals and closed subsets of the spectrum.\nIt gives $\\sqrt{I} = \\bigcap_{P \\supseteq I} P$, so two ideals have the same zero locus in $\\operatorname{Spec} R$ exactly when they have the same radical.\nThe proof is a standard use of Zorn's lemma, in the same style as the [[existence of maximal ideals|02HJYT53F7]].\nOver a polynomial ring over an algebraically closed field, [[Hilbert's Nullstellensatz|D85VZWS9QK]] sharpens it: maximal ideals alone already suffice.\nCompare the [[Jacobson radical|XA0DCNH7WY]], which is the intersection of the maximal ideals only and contains the nilradical.\n",
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