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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]] with unit that is [[Noetherian|0MVH4Y1CRZ]] on both sides, that is, left Noetherian and right Noetherian.\nLet $J$ be the [[Jacobson radical|7HWZ06SRMF]] of $R$.\nThen\n$$\\bigcap_{n \\ge 1} J^n = 0.$$\n\n## Remarks\n\nThe conjecture is open in general.\nFor commutative Noetherian rings it holds, by the Krull intersection theorem.\nThe one-sided version, for rings that are only right Noetherian, is false; I. N. Herstein gave a counterexample.\nThe conjecture is known for several classes of two-sided Noetherian rings, among them the fully bounded Noetherian rings.\n\n## Explanations\n\n### L1 human\n\nThe Jacobson radical $J$ collects the elements of a ring that are \"small\" in a precise sense.\nIts powers $J, J^2, J^3, \\ldots$ shrink, and the conjecture says that in a Noetherian ring nothing survives in all of them.\nFor commutative rings this is a classical theorem of Krull, and the question is whether the same holds without commutativity.\n",
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      "first_stated": "1956",
      "posed_by": "Nathan Jacobson"
    },
    "kind": "conjecture",
    "msc": [
      "16P40",
      "16N20"
    ],
    "name": "Jacobson's conjecture",
    "same_as": {
      "wikidata": "Q16000128",
      "wikipedia": "Jacobson's conjecture"
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    "tag": "BFP0ZST7V3",
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/BFP0ZST7V3/",
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  "markdown": "https://tamnd.github.io/math2/node/BFP0ZST7V3.md",
  "msc": [
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  "name": "Jacobson's conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/BFP/BFP0ZST7V3.md",
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  "tag": "BFP0ZST7V3",
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  "url": "https://tamnd.github.io/math2/wiki/jacobsons-conjecture/",
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