{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nAn *endomorphism* of $M$ is a map $f \\colon M \\to M$ with $f(x + y) = f(x) + f(y)$ and $f(rx) = r f(x)$ for all $x, y \\in M$ and $r \\in R$.\nThe *endomorphism ring* $\\End_R(M)$ is the set of endomorphisms of $M$ with the operations\n$$(f + g)(x) = f(x) + g(x), \\qquad (fg)(x) = f(g(x)).$$\n\n## Remarks\n\n$\\End_R(M)$ is a ring with zero map as zero and the identity map $\\id_M$ as unit element.\nFor an abelian group $A$, viewed as a $\\Z$-module, one writes $\\End(A) = \\End_\\Z(A)$.\nExample: $\\End_\\Z(\\Z/n\\Z) \\cong \\Z/n\\Z$.\nThe check: an endomorphism $f$ is determined by $f(1) = a$, since $f(k) = ka$, and every $a$ gives one; composition corresponds to multiplication of the values $a$.\nNon-example: the map $x \\mapsto x + 1$ on $\\Z$ is not in $\\End_\\Z(\\Z)$, since it does not preserve addition: it sends $0$ to $1$.\nFor a field $K$, choosing a basis gives $\\End_K(K^n) \\cong M_n(K)$.\nFor the ring $R$ as a left module over itself, $\\End_R(R) \\cong R^{\\mathrm{op}}$, the [[opposite ring|M8Q24XB0AF]], through $f \\mapsto f(1)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "AWSBZWWMG9",
    "kind": "definition",
    "msc": [
      "16S50"
    ],
    "name": "Endomorphism ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Endomorphism_ring"
      }
    ],
    "tag": "D5MMVPJ8M1",
    "uses": [
      "0894J980RK",
      "09VDS2W1E8",
      "M8Q24XB0AF"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/D5MMVPJ8M1/",
  "links": [
    {
      "field": "defines",
      "target": "AWSBZWWMG9"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    },
    {
      "field": "uses",
      "target": "09VDS2W1E8"
    },
    {
      "field": "uses",
      "target": "M8Q24XB0AF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/D5MMVPJ8M1.md",
  "msc": [
    "16S50"
  ],
  "name": "Endomorphism ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D5M/D5MMVPJ8M1.md",
  "status": null,
  "tag": "D5MMVPJ8M1",
  "uri": "m2://D5MMVPJ8M1",
  "url": "https://tamnd.github.io/math2/wiki/endomorphism-ring-definition/",
  "writing": {
    "tag": "D5MMVPJ8M1",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}