The Coordinate Ring of an Affine Variety

Recall from the Affine Varieties page that an affine variety is simply an irreducible affine algebraic set. When we have a nonempty affine variety $V$ we can consider what is called the coordinate ring of $V$ which we define below.

Definition: Let $K$ be a field and let $V \subset \mathbb{A}^n(K)$ be an nonempty affine variety. Then the Coordinate Ring of $V$ is the ring $\Gamma(V) = K[x_1, x_2, ..., x_n]/I(V)$.

Here the notation "$K[x_1, x_2, ..., x_n]/I(V)$" means the quotient ring of $K[x_1, x_2, ..., x_n]$ with respect to the ideal of $V$, $I(V)$.

The following proposition tells us that coordinate rings are integral domains.

Proposition 1: Let $K$ be a field and let $V \subset \mathbb{A}^n(K)$ be a nonempty affine variety. Then the coordinate ring $\Gamma(V)$ is an integral domain.

We use the following result from algebra: If $R$ is a ring and $I$ is an ideal in $R$ then $I$ is a prime ideal if and only if $R/I$ is an integral domain.

• Proof: Since $V$ is a nonempty affine variety, the ideal of $V$, $I(V)$ is a prime ideal. But this immediately implies that $\Gamma(V) = K[x_1, x_2, ..., x_n]/I(V)$ is an integral domain. $\blacksquare$