{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.\nThe module $M$ is *finitely generated* when there are $x_1, \\dots, x_n \\in M$ such that each $x \\in M$ is\n$$x = r_1 x_1 + \\cdots + r_n x_n \\qquad \\text{for some } r_1, \\dots, r_n \\in R.$$\n\n## Remarks\n\nOver a field, the finitely generated modules are the finite-dimensional vector spaces.\n$\\Z^n$ and $\\Z / n\\Z$ are finitely generated over $\\Z$; the group $\\Q$ is not.\nOver a Noetherian ring, each submodule of a finitely generated module is finitely generated.\nThe finitely generated modules over a principal ideal domain are classified by the structure theorem.\n\n## Explanations\n\n### L1 human\n\nA module is finitely generated when finitely many elements produce all others by linear combinations.\nThis is the module analogue of a finite-dimensional vector space.\nUnlike in vector spaces, the generators need not be independent.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "05QY1VEBYD",
    "kind": "definition",
    "msc": [
      "13E15",
      "16D10"
    ],
    "name": "Finitely generated module",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Finitely_generated_module"
      }
    ],
    "tag": "0XHM30WBVB",
    "uses": [
      "09VDS2W1E8"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0XHM30WBVB/",
  "links": [
    {
      "field": "defines",
      "target": "05QY1VEBYD"
    },
    {
      "field": "uses",
      "target": "09VDS2W1E8"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0XHM30WBVB.md",
  "msc": [
    "13E15",
    "16D10"
  ],
  "name": "Finitely generated module",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0XH/0XHM30WBVB.md",
  "status": null,
  "tag": "0XHM30WBVB",
  "uri": "m2://0XHM30WBVB",
  "url": "https://math2.tamnd.com/wiki/finitely-generated-module-definition/",
  "writing": {
    "tag": "0XHM30WBVB",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}