Suppose $R$ is a commutative ring, $f\colon F_1\to F_2$ is a homomorphism of free modules, and $M$ is an $R$-module.
If $f$ is a surjective homomorphism, then $f\otimes_R \mathrm{id}_M$ is surjective.
Is it true that if $f$ is a monomorphism, then $f\otimes_R \mathrm{id}_M$ is a monomorphism? I can't think of any counterexamples.