TITLE: Tensor product of domains is a domain
QUESTION [27 upvotes]: I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states:
Let $A$ and $B$ be $k$-algebras, which are also domains, with $k = \overline{k}$ and $A$ finitely generated. Then $A \otimes_k B$ is a domain.
For homework I've proved that the direct product of two irreducible algebraic sets is again an irreducible algebraic set. The teacher simply stated that this implies the above result when $A, B$ are finitely generated. So, my first question is, can someone explain this implication to me?
My second question is, is the above result from Milne true if both $A$ and $B$ are not finitely-generated? If not, can you provide a counterexample?

REPLY [12 votes]: Let $K$ be an algebraically closed field and $A$, $B$ two $K$-algebras which are integral domains. Then $A\otimes_K B$ is an integral domain.

Let $x,x'\in A\otimes_K B$ such that $xx'=0$. Write $x=\sum a_i\otimes b_i$ and $x'=\sum a_j'\otimes b_j'$. By taking minimal representations (as sums of monomial tensors) for $x$ and $y$ one can assume that $(a_i)$, $(a_j')$, $(b_i)$ and $(b_j')$ are linearly independent over $K$. Now considering $A'$ the $K$-subalgebra of $A$ generated by $(a_i)$ and $(a_j')$, and analogously $B'$ the $K$-subalgebra of $B$ generated by $(b_i)$ and $(b_j')$ we reduce the problem to the affine case that is proved in Milne's book.