The Coordinate Ring of an Affine Variety

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\$