Given an affine variety in the -dimensional affine space , where is an algebraically closed field, the coordinate ring of is the quotient ring
where is the ideal formed by all polynomials with coefficients in which are zero at all points of . If is the entire -dimensional affine space , then this ideal is the zero ideal. It follows that the coordinate ring of is the polynomial ring . The coordinate ring of a plane curve defined by the Cartesian equation in the affine plane is .
In general, the Krull dimension of ring is equal to the dimension of as a closed set of the Zariski topology of .
Two polynomials and define the same function on iff . Hence the elements of are equivalence classes which can be identified with the polynomial functions from to .