{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a field, let $n \\ge 0$, and let $X \\subseteq k^n$ be an [[algebraic set|VHFMQQYSTN]] with [[ideal|HH6VEVM6Z0]] $I(X)$.\nThe *coordinate ring* of $X$ is the [[quotient ring|0076HWHGHQ]]\n$$k[X] = k[x_1, \\dots, x_n] / I(X)$$\nof the [[polynomial ring|0SDC934R52]].\nIt is also written $A(X)$ or $\\Gamma(X)$.\n\n## Remarks\n\nTwo polynomials have the same class in $k[X]$ if and only if they define the same function on $X$.\nSo $k[X]$ is the ring of [[polynomial functions|1G0B73WPX8]] $X \\to k$.\nIt is a finitely generated $k$-algebra, generated by the classes of $x_1, \\dots, x_n$.\nIt is a [[reduced ring|SZHY975FFX]], since $I(X)$ is a radical ideal.\n\nExample: for the parabola $X = V(y - x^2)$ over an infinite field, $k[X] = k[x, y]/(y - x^2) \\cong k[x]$, through $x \\mapsto x$ and $y \\mapsto x^2$.\n\nNon-example: the ring $k[x]/(x^2)$ is not the coordinate ring of any algebraic set, because it has the nonzero nilpotent element $x$.\n\n$X$ is irreducible if and only if $k[X]$ is an [[integral domain|0QVD7WBFGN]]; when $k$ is algebraically closed, this says that $X$ is an [[affine variety|JERVQT4C12]].\nIf $k$ is algebraically closed, [[Hilbert's Nullstellensatz|D85VZWS9QK]] shows that the points of $X$ correspond to the [[maximal ideals|0PV74TR5FP]] of $k[X]$.\nThe point $a$ corresponds to the ideal of functions that vanish at $a$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "G9WB27VWP7",
    "kind": "definition",
    "msc": [
      "14A10",
      "14A05"
    ],
    "name": "Coordinate ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Affine_variety"
      }
    ],
    "tag": "DZWS3QN3H7",
    "uses": [
      "VHFMQQYSTN",
      "HH6VEVM6Z0",
      "0SDC934R52",
      "0076HWHGHQ",
      "1G0B73WPX8",
      "JERVQT4C12",
      "0QVD7WBFGN",
      "SZHY975FFX",
      "0PV74TR5FP",
      "D85VZWS9QK"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/DZWS3QN3H7/",
  "links": [
    {
      "field": "defines",
      "target": "G9WB27VWP7"
    },
    {
      "field": "uses",
      "target": "VHFMQQYSTN"
    },
    {
      "field": "uses",
      "target": "HH6VEVM6Z0"
    },
    {
      "field": "uses",
      "target": "0SDC934R52"
    },
    {
      "field": "uses",
      "target": "0076HWHGHQ"
    },
    {
      "field": "uses",
      "target": "1G0B73WPX8"
    },
    {
      "field": "uses",
      "target": "JERVQT4C12"
    },
    {
      "field": "uses",
      "target": "0QVD7WBFGN"
    },
    {
      "field": "uses",
      "target": "SZHY975FFX"
    },
    {
      "field": "uses",
      "target": "0PV74TR5FP"
    },
    {
      "field": "uses",
      "target": "D85VZWS9QK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DZWS3QN3H7.md",
  "msc": [
    "14A10",
    "14A05"
  ],
  "name": "Coordinate ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DZW/DZWS3QN3H7.md",
  "status": null,
  "tag": "DZWS3QN3H7",
  "uri": "m2://DZWS3QN3H7",
  "url": "https://tamnd.github.io/math2/wiki/coordinate-ring-definition/",
  "writing": {
    "tag": "DZWS3QN3H7",
    "kind": "definition",
    "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
      }
    ]
  }
}