I've been going through my submitted exercises again of my Commutative Algebra-class and I have the following question:
Let $A$ be a commutative ring with unity. Given any injective homomorphism of $B$-modules $M \rightarrow M'$, where $B$ is any $A$-algebra, is it necessarily true that $M \rightarrow M'$ is injective when considered as an $A$-linear map? In the solution I handed in, I used the rather weak argument that "injectivity is a set-theoretic property, so stays preserved" - which wasn't marked false by the assistant.
However, the reasoning doesn't quite convince me. Is it true that injectivity stays preserved? If not, is there any counterexample?
Thanks a lot!