coordinate ring

If a variety is a shape, its coordinate ring is the collection of all polynomial 'measurements' you can make on that shape. Two polynomials that agree at every point of the variety are the same measurement, so the coordinate ring is the polynomial ring with that redundancy quotiented away. It packages all the geometry into a single algebraic object.

Precisely, for an affine variety X in A^n the coordinate ring is k[X] = k[x_1, ..., x_n] / I(X), where I(X) is the ideal of all polynomials vanishing on X. Its elements are the regular (polynomial) functions on X, and over an algebraically closed field its maximal ideals correspond bijectively to the points of X by the Nullstellensatz. This is the heart of the algebra-geometry dictionary: morphisms of varieties X to Y correspond contravariantly to k-algebra homomorphisms k[Y] to k[X].

The properties of the ring read off the geometry of the variety. The ring k[X] is reduced (no nonzero nilpotents) exactly because I(X) is a radical ideal; it is an integral domain exactly when X is irreducible; and its Krull dimension equals the dimension of X. A finitely generated reduced k-algebra is the same data as an affine variety over k — the abstraction that the affine-scheme viewpoint later completes by dropping the reducedness assumption.