Is an injective map of $B$-modules also injective as an $A$-linear map if $B$ is an $A$-algebra?

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!

• A map being an injection simply has nothing to do with the module structure, and that is essentially what you said. – rschwieb Dec 30 '15 at 14:04

1 Answer

Your argument is correct. Injectivity is a set-theoretic property: it says that if two elements $m_1$ and $m_2$ of $M$ satisfy $f\left(m_1\right) = f\left(m_2\right)$ (where $f$ is your map), then $m_1 = m_2$. There is nothing here that would depend on whether $M$ and $M'$ are considered as $B$-modules, as $A$-modules, or as sets. Thus, whether $M$ and $M'$ are considered as $B$-modules, as $A$-modules, or as sets does not matter for injectivity of $f$.

Your doubts suggest that you might be mistaking injectivity for the categorical notion of monomorphism (which would still be independent on whether $M$ and $M'$ are considered as $B$-modules, as $A$-modules, or as sets; but at least this independency would not be totally obvious, because the definition of a monomorphism involves the category).