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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nThe ring $R$ is a *Jacobson ring* if every [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$ is the intersection of the [[maximal ideals|0PV74TR5FP]] that contain $\\mathfrak{p}$.\n\n## Remarks\n\nExample: $\\Z$ is a Jacobson ring.\nA nonzero prime ideal $p\\Z$ is maximal, and the prime $0$ equals $\\bigcap_p p\\Z$, because a nonzero integer has only finitely many prime divisors.\n\nNon-example: the localization $\\Z_{(p)}$ is not a Jacobson ring.\nIts only maximal ideal is $p\\Z_{(p)}$, so the intersection of the maximal ideals containing the prime $0$ is $p\\Z_{(p)} \\ne 0$.\n\nEquivalent forms: $R$ is Jacobson if and only if every [[radical ideal|437YRCNYRJ]] is an intersection of maximal ideals.\nIt is also equivalent that for every ideal $I$, the [[nilradical|QQDN6EYQNJ]] of $R/I$ equals its [[Jacobson radical|XA0DCNH7WY]].\nSome texts use the name Hilbert ring.\n",
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