Proposition 2

— Let $A$ and $B$ be two algebras over a field $K$. If $A$ is an integral domain and $B$ is regular then $A \otimes_K B$ is an integral domain.

Let $L$ be the field of fractions of $A$. Since $B$ is a regular $K$-algebra, $L \otimes_K B$ is an integral domain, hence so is $A \otimes_K B$, since it is isomorphic to a subring of $L \otimes_K B$.