{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a field and let $G$ be a [[torsion-free group|PEMNN32A43]].\nThat is, every element of $G$ other than the identity has infinite order.\nLet $K[G]$ be the [[group ring|4NMTSX0XDS]] of $G$ over $K$.\n\n1. (Zero divisor conjecture) $K[G]$ has no [[zero divisors|0NNRDW10RZ]] other than $0$; see [[Kaplansky's zero divisor conjecture|E1NE8V7EX8]].\n2. (Idempotent conjecture) The only [[idempotents|0N5NJZKKN3]] of $K[G]$ are $0$ and $1$.\n3. (Unit conjecture) Every [[unit|0XVEZDE9J8]] of $K[G]$ has the form $c g$ with $c \\in K$ nonzero and $g \\in G$.\n\n## Remarks\n\nThe conjectures are linked: the unit conjecture implies the zero divisor conjecture, and the zero divisor conjecture implies the idempotent conjecture, since an idempotent $e$ satisfies $e(1 - e) = 0$.\nThe hypothesis that $G$ is torsion-free is needed: if $g \\ne 1$ has finite order $n$, then $(1 - g)(1 + g + \\cdots + g^{n-1}) = 1 - g^n = 0$.\nThe unit conjecture goes back to the 1940 thesis of Graham Higman.\nIn 2021 Giles Gardam disproved the unit conjecture over the field with two elements, with a nontrivial unit in the group ring of the fundamental group of the Hantzsche–Wendt manifold.\nAll three conjectures hold for groups with the unique product property, and the zero divisor conjecture holds for torsion-free elementary amenable groups.\nThe zero divisor and idempotent conjectures are open in general.\nKaplansky also made conjectures in other fields, on Banach algebras and on quadratic forms; they are separate questions.\n\n## Explanations\n\n### L1 human\n\nThe group ring of a group mixes the group with a field, and elements of finite order in the group always create zero divisors and idempotents in it.\nKaplansky asked whether these are the only sources: if the group has no elements of finite order, is the group ring as well behaved as a ring of polynomials?\nThree precise forms of the question concern zero divisors, idempotents and units.\nThe one about units turned out to be false, and the other two are still open.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "4NMTSX0XDS"
    ],
    "history": {
      "posed_by": "Irving Kaplansky"
    },
    "kind": "conjecture",
    "msc": [
      "16S34",
      "16U60",
      "16U40"
    ],
    "name": "Kaplansky's conjectures",
    "same_as": {
      "wikidata": "Q6366842",
      "wikipedia": "Kaplansky's conjectures"
    },
    "sources": [
      {
        "locator": "Group rings",
        "url": "https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures"
      }
    ],
    "tag": "DSF8C6QPAG",
    "uses": [
      "4NMTSX0XDS",
      "PEMNN32A43",
      "0NNRDW10RZ",
      "0N5NJZKKN3",
      "0XVEZDE9J8",
      "E1NE8V7EX8"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/DSF8C6QPAG/",
  "links": [
    {
      "field": "about",
      "target": "4NMTSX0XDS"
    },
    {
      "field": "uses",
      "target": "4NMTSX0XDS"
    },
    {
      "field": "uses",
      "target": "PEMNN32A43"
    },
    {
      "field": "uses",
      "target": "0NNRDW10RZ"
    },
    {
      "field": "uses",
      "target": "0N5NJZKKN3"
    },
    {
      "field": "uses",
      "target": "0XVEZDE9J8"
    },
    {
      "field": "uses",
      "target": "E1NE8V7EX8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DSF8C6QPAG.md",
  "msc": [
    "16S34",
    "16U60",
    "16U40"
  ],
  "name": "Kaplansky's conjectures",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DSF/DSF8C6QPAG.md",
  "status": null,
  "tag": "DSF8C6QPAG",
  "uri": "m2://DSF8C6QPAG",
  "url": "https://tamnd.github.io/math2/wiki/kaplanskys-conjectures/",
  "writing": {
    "tag": "DSF8C6QPAG",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}