{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with $1$.\nThe ring $R$ is *Dedekind-finite* if, for all $a, b \\in R$, the equation $ab = 1$ implies $ba = 1$.\n\n## Remarks\n\nExample: for a field $F$ and $n \\ge 1$, the [[matrix ring|0FHVPRAE60]] $M_n(F)$ is Dedekind-finite.\nIndeed, if $AB = I$, then $\\det A \\det B = 1$, so $A$ is invertible, and $B = A^{-1}(AB) = A^{-1}$, hence $BA = I$.\n\nNon-example: let $V$ be the vector space of sequences $(x_1, x_2, \\dots)$ over a field and $R = \\End(V)$.\nLet $a(x_1, x_2, \\dots) = (x_2, x_3, \\dots)$ and $b(x_1, x_2, \\dots) = (0, x_1, x_2, \\dots)$.\nThen $ab = 1$, but $ba(x_1, x_2, \\dots) = (0, x_2, x_3, \\dots)$, so $ba \\ne 1$.\n\nOther names are *directly finite* and *von Neumann finite*.\nEquivalently, $R$ is not isomorphic as a left $R$-module to $R \\oplus N$ for any nonzero module $N$.\nEvery commutative ring, every finite ring, and every left Noetherian ring is Dedekind-finite.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GV1HKEAWVN",
    "kind": "definition",
    "msc": [
      "16U99",
      "16U60"
    ],
    "name": "Dedekind-finite ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dedekind-finite_ring"
      }
    ],
    "tag": "5VYYQER2JB",
    "uses": [
      "0B1261B5ED",
      "0FHVPRAE60"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/5VYYQER2JB/",
  "links": [
    {
      "field": "defines",
      "target": "GV1HKEAWVN"
    },
    {
      "field": "uses",
      "target": "0B1261B5ED"
    },
    {
      "field": "uses",
      "target": "0FHVPRAE60"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5VYYQER2JB.md",
  "msc": [
    "16U99",
    "16U60"
  ],
  "name": "Dedekind-finite ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5VY/5VYYQER2JB.md",
  "status": null,
  "tag": "5VYYQER2JB",
  "uri": "m2://5VYYQER2JB",
  "url": "https://tamnd.github.io/math2/wiki/dedekind-finite-ring-definition/",
  "writing": {
    "tag": "5VYYQER2JB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}