{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "EK0T7FRKCY"
    }
  ],
  "body": "\n## Statement\n\nLet $R$ be a ring and let $M$ be a [[simple|G00C22AEDH]] left $R$-module.\nThen the [[endomorphism ring|AWSBZWWMG9]] $\\End_R(M)$ is a [[division ring|0EA5BTA0CX]].\n\n## Remarks\n\n$\\End_R(M)$ is nonzero, since $M \\ne 0$ gives $\\id_M \\ne 0$.\nBy [[Schur's lemma|X4M2FFKQJ6]] every nonzero endomorphism of $M$ is an isomorphism, so its inverse map is again an endomorphism and is its inverse in $\\End_R(M)$.\nExample: $\\End_\\Z(\\Z/p\\Z) \\cong \\Z/p\\Z$, a field, for $p$ prime.\nSimplicity is needed: for the semisimple $\\Z$-module $M = \\Z/p\\Z \\oplus \\Z/p\\Z$, the ring $\\End_\\Z(M) \\cong M_2(\\Z/p\\Z)$ has the nonzero non-invertible element $E_{11}$.\nIf $R$ is an algebra over an algebraically closed field $k$ and $M$ has finite dimension over $k$, then $\\End_R(M) = k$.\n\n## Explanations\n\n### L1 human\n\nThe self-maps of a simple module that are not zero can always be undone.\nSo the self-maps form a ring in which one can divide by every nonzero element, just as in a field, except that the order of multiplication may matter.\n\n### L2 human\n\nThis is how division rings enter the structure theory of rings.\nIn the Artin–Wedderburn theorem, a simple Artinian ring is a matrix ring over the division ring $\\End_R(M)^{\\mathrm{op}}$ for a simple module $M$.\nIn representation theory over $\\C$, it says that endomorphisms of a finite-dimensional irreducible representation are scalars.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "AWSBZWWMG9",
      "G00C22AEDH"
    ],
    "kind": "statement",
    "msc": [
      "16D60",
      "16S50"
    ],
    "name": "Endomorphism ring of a simple module is a division ring",
    "role": "corollary",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schur%27s_lemma"
      }
    ],
    "tag": "TRMYSF91EV",
    "uses": [
      "G00C22AEDH",
      "AWSBZWWMG9",
      "0EA5BTA0CX",
      "X4M2FFKQJ6"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/TRMYSF91EV/",
  "links": [
    {
      "field": "about",
      "target": "AWSBZWWMG9"
    },
    {
      "field": "about",
      "target": "G00C22AEDH"
    },
    {
      "field": "uses",
      "target": "G00C22AEDH"
    },
    {
      "field": "uses",
      "target": "AWSBZWWMG9"
    },
    {
      "field": "uses",
      "target": "0EA5BTA0CX"
    },
    {
      "field": "uses",
      "target": "X4M2FFKQJ6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TRMYSF91EV.md",
  "msc": [
    "16D60",
    "16S50"
  ],
  "name": "Endomorphism ring of a simple module is a division ring",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TRM/TRMYSF91EV.md",
  "status": null,
  "tag": "TRMYSF91EV",
  "uri": "m2://TRMYSF91EV",
  "url": "https://tamnd.github.io/math2/wiki/endomorphism-ring-of-a-simple-module-is-a-division-ring/",
  "writing": {
    "tag": "TRMYSF91EV",
    "kind": "statement",
    "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
      }
    ]
  }
}