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\$