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  "body": "\n## Summary\n\nA Jacobson ring, also called a Hilbert ring, is a [[commutative ring|0310REJJY8]] in which every [[prime ideal|00MHHZ9SNE]] is an intersection of [[maximal ideals|0PV74TR5FP]].\nIn such a ring, the maximal ideals alone determine all radical ideals, as in [[Hilbert's Nullstellensatz|D85VZWS9QK]].\nFields, $\\Z$, and every finitely generated algebra over a Jacobson ring are Jacobson rings; this last fact is a general form of the Nullstellensatz.\nIn geometry, the closed points of the spectrum of a Jacobson ring are dense in every closed subset.\n\n## Explanations\n\n### L1 human\n\nIn the ring of polynomials over $\\C$, a curve is determined by its points, and each point is a maximal ideal.\nA Jacobson ring is a ring where every prime ideal is determined in this way by the maximal ideals above it.\nThe integers are an example: the prime ideal $0$ is the intersection of all ideals $p\\Z$, since no nonzero integer is divisible by every prime.\nA local domain that is not a field, such as $\\Z_{(p)}$, is not Jacobson.\nIts only maximal ideal is $p\\Z_{(p)}$, and this ideal cannot recover the prime $0$.\n",
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