product has coordinate ring . Let be an algebraic subset of and an algebraic subset of Then each is a polynomial in , and each is in . The product of and is defined as the algebraic set in The product is irreducible if each , is irreducible. It is important to note that the Zariski topology on is not the topological product of the Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets and Hence, polynomials that are in but cannot be obtained as a product of a polynomial in with a polynomial in will define algebraic sets that are in the Zariski topology on but not in the product topology. Morphisms of affine varieties A morphism, or regular map, of affine varieties is a function between affine varieties which is polynomial in each coordinate: more precisely, for affine varieties and , a morphism from to is a map of the form where for each These are the morphisms in the category of affine varieties. There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field and homomorphisms of coordinate rings of affine varieties over going in the opposite direction. Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over and their coordinate rings, the category of affine varieties over is dual to the category of coordinate rings of affine varieties over The category of coordinate rings of affine varieties over is precisely the category of finitely-generated, nilpotent-free algebras over More precisely, for each morphism of affine varieties, there is a homomorphism between the coordinate rings (going in the opposite direction), and for each such homomorphism, there is a morphism of the varieties associated to the coordinate rings. This can be shown explicitly: let and be affine varieties with coordinate rings and respectively. Let be a morphism. Indeed, a homomorphism between polynomial rings factors uniquely through the ring and a homomorphism is determined uniquely by the images of Hence, each homomorphism corresponds uniquely to a choice of image for each . Then given any morphism from to a homomorphism can be constructed which sends to where is the equivalence class of in Similarly, for each homomorphism of the coordinate rings, a morphism of the affine varieties can be constructed in the opposite direction. Mirroring the paragraph above, a