---
tag: DZWS3QN3H7
kind: definition
name: Coordinate ring
msc: [14A10, 14A05]
defines: G9WB27VWP7
uses: [VHFMQQYSTN, HH6VEVM6Z0, 0SDC934R52, 0076HWHGHQ, 1G0B73WPX8, JERVQT4C12, 0QVD7WBFGN, SZHY975FFX, 0PV74TR5FP, D85VZWS9QK]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_variety
---

## Definition

Let $k$ be a field, let $n \ge 0$, and let $X \subseteq k^n$ be an [[algebraic set|VHFMQQYSTN]] with [[ideal|HH6VEVM6Z0]] $I(X)$.
The *coordinate ring* of $X$ is the [[quotient ring|0076HWHGHQ]]
$$k[X] = k[x_1, \dots, x_n] / I(X)$$
of the [[polynomial ring|0SDC934R52]].
It is also written $A(X)$ or $\Gamma(X)$.

## Remarks

Two polynomials have the same class in $k[X]$ if and only if they define the same function on $X$.
So $k[X]$ is the ring of [[polynomial functions|1G0B73WPX8]] $X \to k$.
It is a finitely generated $k$-algebra, generated by the classes of $x_1, \dots, x_n$.
It is a [[reduced ring|SZHY975FFX]], since $I(X)$ is a radical ideal.

Example: 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 $y \mapsto x^2$.

Non-example: the ring $k[x]/(x^2)$ is not the coordinate ring of any algebraic set, because it has the nonzero nilpotent element $x$.

$X$ 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]].
If $k$ is algebraically closed, [[Hilbert's Nullstellensatz|D85VZWS9QK]] shows that the points of $X$ correspond to the [[maximal ideals|0PV74TR5FP]] of $k[X]$.
The point $a$ corresponds to the ideal of functions that vanish at $a$.