coordinate ring

《Introduction to Commutative Algebra》 · 第 16 页 · Chapter 1，Exercise 27

$P(X)$ is generated as a $k$-algebra by the coordinate functions, and is called the coordinate ring (or affine algebra) of $X$.

《Solitons: Differential equations, symmetries, and infinite-dimensional algebras》 · 第 18 页 · Definition 1.30

Let $X\subseteq\mathbb A^n(k)$ be an affine algebraic set. The $k$-algebra $\Gamma(X):=k[T_1,\ldots,T_n]/I(X)\cong\operatorname{Hom}(X,\mathbb A^1(k))$ is called the affine coordinate ring of $X$.