Given an affine variety V in the n-dimensional affine space where K is an algebraically closed field, the coordinate ring of V is the quotient ring

where I(V) is the ideal formed by all polynomials with coefficients in K which are zero at all points of V. If V is the entire n-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 V as a closed set of the Zariski topology of .
Two polynomials and define the same function on V iff . Hence the elements of are equivalence classes which can be identified with the polynomial functions from V to K.