{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ and $S$ be [[rings|0894J980RK]].\nAn *$(R, S)$-bimodule* is an abelian group $M$ with these structures:\n\n1. a left [[$R$-module|09VDS2W1E8]] structure $(r, m) \\mapsto rm$;\n2. a right $S$-module structure $(m, s) \\mapsto ms$;\n3. the compatibility $(rm)s = r(ms)$ for all $r \\in R$, $m \\in M$ and $s \\in S$.\n\n## Remarks\n\nAn $(R, R)$-bimodule is also called an $R$-bimodule.\nExample: for a field $K$, the space $M_{m \\times n}(K)$ of $m \\times n$ matrices is an $(M_m(K), M_n(K))$-bimodule under matrix multiplication.\nClauses 1 and 2 are the usual rules of matrix arithmetic, and clause 3 is associativity $(AX)B = A(XB)$.\nEvery ring $R$ is an $R$-bimodule, and its sub-bimodules are its two-sided ideals.\nNon-example: let $R$ be a noncommutative ring and give $R$ the \"right action\" $m \\cdot s = sm$.\nThis fails clause 2, since $(m \\cdot s) \\cdot t = tsm$ while $m \\cdot (st) = stm$, and $ts \\ne st$ for some $s, t$.\nFor a commutative ring $K$, every $K$-module is a $K$-bimodule with $mk = km$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "EM7FHAFD3N",
    "kind": "definition",
    "msc": [
      "16D20"
    ],
    "name": "Bimodule",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bimodule"
      }
    ],
    "tag": "RBPR7BK5T5",
    "uses": [
      "0894J980RK",
      "09VDS2W1E8"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/RBPR7BK5T5/",
  "links": [
    {
      "field": "defines",
      "target": "EM7FHAFD3N"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    },
    {
      "field": "uses",
      "target": "09VDS2W1E8"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RBPR7BK5T5.md",
  "msc": [
    "16D20"
  ],
  "name": "Bimodule",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RBP/RBPR7BK5T5.md",
  "status": null,
  "tag": "RBPR7BK5T5",
  "uri": "m2://RBPR7BK5T5",
  "url": "https://tamnd.github.io/math2/wiki/bimodule-definition/",
  "writing": {
    "tag": "RBPR7BK5T5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}