TITLE: Examples for the fact that a pullback of an epimorphism is not necessarily an epimorphism.
QUESTION [6 upvotes]: I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a counterexample in the category Haus.
I was wondering whether there also exist easy counterexamples in Ring, the category of commutative rings with a 1 element (or in Grp, Rng, etc.). I have been playing around with the embedding of $\mathbb{Z}$ in $\mathbb{Q}$, which is an epimorphism, but I couldn't find anything.

REPLY [5 votes]: An example in $\mathsf{Ring}$ is the inclusion $\mathbb{C}[t]\to\mathbb{C}[t,t^{-1}]$, which is an epimorphism, but whose pullback along $\mathbb{C}[t^{-1}]\to\mathbb{C}[t,t^{-1}]$ is $\mathbb{C}\to\mathbb{C}[t^{-1}]$, which is not an epimorphism.
However, every pullback of $\mathbb{Z}\to\mathbb{Q}$ is an epimorphism. There's a proof in the answer to this related question.