Products and the Hausdorff Axiom

We want to define the product X x Y of any two prevarieties X,Y. Now we will certainly want to have An x Am = An+m. But the product of the Zariski topologies in An and Am does not give the Zariski topology in An+m in A1 x A1, for instance, the only closed sets in the product topology are finite unions of horizontal and vertical lines. The only reliable way to find the correct definition is to use the general category-theoretic definition of product. 

Definition 1. Let C be a category, X, Y objects in C. An object Z plus two morphisms 

We shall prove that products exist in the category of prevarieties over k. Note that we have no choice for the underlying set for if X x Y is a product of the prevarieties X and Y, X x Y as a point set must be the usual product of the point sets X and Y. To see this, let W be a simple point this is a prevariety (A0, in fact). The maps of W to any prevariety S clearly correspond to the points of S, and by definition hom(W,X x7) hom(W,X) x hom(V,F). 

Proposition 1. Let X and Y be affine varieties, with coordinate rings R and S. Then 

Proof We recall the following result from commutative algebra let R and S be integral domains over the algebraically closed field k. Then R S kS is an integral domain. [Cf. Zariski-Samuel, vol. 1, Ch. 3, 15.] 

---