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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]].\nA one-sided ideal of $R$ is *nil* if each of its elements is nilpotent.\nThe conjecture states that if $R$ has no nonzero [[nil ideal|NZ1G9WRWFB]] (two-sided), then $R$ has no nonzero nil [[left ideal|5N2CJZ47D4]].\n\n## Remarks\n\nThe conjecture is open.\nEquivalent forms: in every ring the sum of two nil left ideals is nil; in every ring every nil left ideal is contained in the [[upper nilradical|3G2JQEYBEK]].\nKrempa showed that it is equivalent to the statement that for every nil ring $R$ the polynomial ring $R[x]$ is equal to its Jacobson radical.\nIt holds for left Noetherian rings, where every nil one-sided ideal is nilpotent by [[Levitzki's theorem|QHD4KJYFSA]].\nSmoktunowicz constructed a nil ring $R$ such that $R[x]$ is not nil, which rules out a stronger form that had been suggested.\n\n## Explanations\n\n### L1 human\n\nAn element is nilpotent when some power of it is zero.\nIn a noncommutative ring there are several ways to collect nilpotent elements: in two-sided ideals or in one-sided ideals.\nThe conjecture says that one-sided ideals of nilpotent elements can only exist if a two-sided one does.\nIt is one of the oldest open problems of ring theory.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "3G2JQEYBEK"
    ],
    "history": {
      "first_stated": "1930",
      "posed_by": "Gottfried Köthe"
    },
    "kind": "conjecture",
    "msc": [
      "16N40"
    ],
    "name": "Köthe conjecture",
    "same_as": {
      "wikidata": "Q6454025",
      "wikipedia": "Köthe conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/K%C3%B6the_conjecture"
      }
    ],
    "tag": "RT2D9JEVRR",
    "uses": [
      "0894J980RK",
      "5N2CJZ47D4",
      "NZ1G9WRWFB",
      "3G2JQEYBEK"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/RT2D9JEVRR/",
  "links": [
    {
      "field": "about",
      "target": "3G2JQEYBEK"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    },
    {
      "field": "uses",
      "target": "5N2CJZ47D4"
    },
    {
      "field": "uses",
      "target": "NZ1G9WRWFB"
    },
    {
      "field": "uses",
      "target": "3G2JQEYBEK"
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  ],
  "markdown": "https://tamnd.github.io/math2/node/RT2D9JEVRR.md",
  "msc": [
    "16N40"
  ],
  "name": "Köthe conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RT2/RT2D9JEVRR.md",
  "status": null,
  "tag": "RT2D9JEVRR",
  "uri": "m2://RT2D9JEVRR",
  "url": "https://tamnd.github.io/math2/wiki/kothe-conjecture/",
  "writing": {
    "tag": "RT2D9JEVRR",
    "kind": "conjecture",
    "level": "W0",
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        "note": "the row of the inventory that the node accounts for",
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