{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3Z7YMK2WDS"
    },
    {
      "field": "uses",
      "target": "4MBR0W0S6N"
    },
    {
      "field": "uses",
      "target": "7Z28JY8QW4"
    },
    {
      "field": "uses",
      "target": "HCD6W2XK6W"
    },
    {
      "field": "uses",
      "target": "PYABBA1QQE"
    },
    {
      "field": "uses",
      "target": "RASAC44RTQ"
    },
    {
      "field": "uses",
      "target": "Y2XHZKR0TX"
    },
    {
      "field": "uses",
      "target": "YBJ2VQHFDQ"
    },
    {
      "field": "about",
      "target": "4MBR0W0S6N"
    },
    {
      "field": "about",
      "target": "7Z28JY8QW4"
    },
    {
      "field": "about",
      "target": "HCD6W2XK6W"
    },
    {
      "field": "about",
      "target": "YBJ2VQHFDQ"
    },
    {
      "field": "specializes",
      "target": "764GZHTGV8"
    },
    {
      "field": "defines",
      "target": "QM5ZCBJYP1"
    }
  ],
  "body": "\n## Summary\n\nAn [[algebra|DPBJ04JMH8]] $B$ over a commutative ring $A$ is of finite type if finitely many elements of $B$ generate it as an $A$-algebra.\nEquivalently, $B$ is a quotient of a [[polynomial ring|0SDC934R52]] $A[x_1, \\ldots, x_n]$ in finitely many variables.\nFinitely generated algebras over a field are the coordinate rings of affine varieties, the main objects of classical algebraic geometry.\nOver a Noetherian base they are Noetherian, by [[Hilbert's basis theorem|0YKA9J3XEE]].\n\n## Explanations\n\n### L1 human\n\nAn algebra is of finite type if every element can be written as a polynomial, with coefficients in $A$, in a fixed finite list of elements.\nThe ring $\\C[x, y]/(x^2 + y^2 - 1)$ of polynomial functions on a circle is generated over $\\C$ by the two coordinate functions.\nFinite type is weaker than finite: $k[x]$ is of finite type over $k$, generated by $x$, but it is not a finitely generated $k$-module.\nThe ring $\\Q$ is not of finite type over $\\Z$: finitely many fractions only produce finitely many prime denominators.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13E15",
      "13B02"
    ],
    "name": "Finite type algebra",
    "same_as": {
      "wikidata": "Q2835968",
      "wikipedia": "Finitely generated algebra"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Finitely_generated_algebra"
      }
    ],
    "specializes": [
      "DPBJ04JMH8"
    ],
    "tag": "AC5TFFQT9M"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/AC5TFFQT9M/",
  "links": [
    {
      "field": "specializes",
      "target": "DPBJ04JMH8"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AC5TFFQT9M.md",
  "msc": [
    "13E15",
    "13B02"
  ],
  "name": "Finite type algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AC5/AC5TFFQT9M.md",
  "status": null,
  "tag": "AC5TFFQT9M",
  "uri": "m2://AC5TFFQT9M",
  "url": "https://math2.tamnd.com/wiki/finite-type-algebra/",
  "writing": {
    "tag": "AC5TFFQT9M",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}