Exercise 30, § 7

Let $A$ be an integral domain and $E, F$ two $A$-modules. Show that if $x$ is a free element of $E$ and $y$ a free element of $F$, then $x \otimes y \neq 0$ in $E \otimes_A F$. (Reduce it to the case where $E$ and $F$ are torsion-free, then to the case where $E$ and $F$ are finitely generated and use Exercise 29 (b) to show that there is an $A$-bilinear mapping $f$ of $E \times F$ into $K$ such that $f(x, y) \neq 0$.)