Exercise 34, § 7

Let $A$ be an integral domain, $P$ a projective $A$-module, $P'$ a projective submodule of $P$ and $j : P' \to P$ the canonical injection. Show that for every torsion-free $A$-module $E$, the homomorphism $j \otimes 1 : P' \otimes_A E \to P \otimes_A E$ is injective (reduce it to the case where $E$ is finitely generated and embed $E$ in a free $A$-module).