Lemma 4

Let $A$ and $B$ be two integral k-algebras, $M$ an $A$-module without torsion, $N$ a $B$-module without torsion. If the ring $A \otimes_k B$ is integral, then $M \otimes_k N$ is a module without torsion over $A \otimes_k B$.

Let $K$ (resp. $L$) be the field of fractions of $A$ (resp. $B$). There exists a set $I$ (resp. $J$) such that $M$ (resp. $N$) is isomorphic to a submodule of $K^{(I)}$ (resp. $L^{(J)}$). The $(A \otimes_k B)$-module $M \otimes_k N$ is therefore isomorphic to a submodule of $K^{(I)} \otimes_k L^{(J)}$, which is isomorphic to $(K \otimes_k L)^{(I \times J)}$. Since $K \otimes_k L$ is a ring of fractions of the integral ring $A \otimes_k B$, it is a torsion-free module over $A \otimes_k B$, whence the lemma.