Pushout of a mono which is not a mono.

Question

I'm studying abelian categories in F. Borceux. I'm trying to solve an exercise : prove that monomorphisms are not stable under pushouts in the category of commutative rings. Any help?

Answer 1

This is the counterexample I found with the previous help :

We use the monomorphism $g : \mathbb{Z} \rightarrow \mathbb{Q}$ and the morphism $f : \mathbb{Z} \rightarrow \mathbb{Z}_n$ with $n > 1$. The pushout $g'$ of $g$ will go from $\mathbb{Z}_n$ to $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}_n$, but this last ring is zero and then $g'$ is not a monomorphism.

This question was motivated by the fact that epimorphisms are always stable under pushout. In abelian categories we also have that monomorphisms are stable under pushout, so I wanted to find a counterexample to make it clear that it is not general.

Answer 2

Hint: Try using the monomorphism $\mathbb{Z}\to\mathbb{Q}$.

A stronger hint is hidden below.

What happens when you push that out along a map $\mathbb{Z}\to C$ where $C$ has torsion?