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  "body": "\n## Statement\n\nLet $K$ be a field and let $G$ be a [[torsion-free group|PEMNN32A43]].\nThen the [[group ring|4NMTSX0XDS]] $K[G]$ has no [[zero divisors|0NNRDW10RZ]] other than $0$: if $a, b \\in K[G]$ and $ab = 0$, then $a = 0$ or $b = 0$.\nIn other words, $K[G]$ is a [[domain|0Q4CBD3HX0]].\n\n## Remarks\n\nThe conjecture is open.\nTorsion-freeness is necessary: if $g \\in G$ has finite order $n \u003e 1$, then $(1 - g)(1 + g + \\cdots + g^{n-1}) = 1 - g^n = 0$ with both factors nonzero.\nIt holds when $G$ has the unique product property: for nonempty finite sets $A, B \\subseteq G$, some product $ab$ with $a \\in A$, $b \\in B$ arises in only one way.\nFor such $G$, take $A$ and $B$ to be the supports of $a$ and $b$.\nThe coefficient of that unique product in $ab$ is a product of two nonzero field elements, so $ab \\ne 0$.\nLeft-orderable groups, and in particular torsion-free abelian groups and free groups, have this property.\nFor $G = \\Z$, the group ring $K[\\Z]$ is the Laurent polynomial ring $K[x, x^{-1}]$, which is a domain.\nThe conjecture is also known for torsion-free elementary amenable groups, whose group rings are Ore domains.\nIn characteristic zero it follows from the strong Atiyah conjecture for $G$.\n\n## Explanations\n\n### L1 human\n\nIn the ring of polynomials, a product of two nonzero polynomials is never zero, because the top degree terms multiply to a nonzero term.\nThe conjecture says the same happens for any group without elements of finite order.\nFor groups that can be ordered there is a \"top term\" and the polynomial argument works.\nFor general torsion-free groups no such argument is known.\n",
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    "about": [
      "4NMTSX0XDS"
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    "history": {
      "posed_by": "Irving Kaplansky"
    },
    "kind": "conjecture",
    "msc": [
      "16S34",
      "16U10"
    ],
    "name": "Kaplansky's zero divisor conjecture",
    "sources": [
      {
        "locator": "Group rings",
        "url": "https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures"
      }
    ],
    "tag": "E1NE8V7EX8",
    "uses": [
      "4NMTSX0XDS",
      "PEMNN32A43",
      "0NNRDW10RZ",
      "0Q4CBD3HX0"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/E1NE8V7EX8/",
  "links": [
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      "field": "uses",
      "target": "PEMNN32A43"
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      "field": "uses",
      "target": "0NNRDW10RZ"
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    {
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      "target": "0Q4CBD3HX0"
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  ],
  "markdown": "https://tamnd.github.io/math2/node/E1NE8V7EX8.md",
  "msc": [
    "16S34",
    "16U10"
  ],
  "name": "Kaplansky's zero divisor conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E1N/E1NE8V7EX8.md",
  "status": null,
  "tag": "E1NE8V7EX8",
  "uri": "m2://E1NE8V7EX8",
  "url": "https://tamnd.github.io/math2/wiki/kaplanskys-zero-divisor-conjecture/",
  "writing": {
    "tag": "E1NE8V7EX8",
    "kind": "conjecture",
    "level": "W0",
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