3
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ A map being an injection simply has nothing to do with the module structure, and that is essentially what you said. $\endgroup$ – rschwieb Dec 30 '15 at 14:04
3
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.