Definition:Affine Coordinate Ring

Definition
Let $k$ be a field.

Let $Y \subseteq k^n$ be an affine algebraic set.

Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.

Let $\map I Y \subseteq k \sqbrk {X_1, \ldots, X_n}$ be the vanishing ideal of $Y$.

Then the affine coordinate ring of $Y$ is defined as the quotient ring:
 * $\ds \map A Y := k \sqbrk {X_1, \ldots, X_n}/\map I Y$