Exercise 10.3.0.5. Let $f: R \rightarrow S$ be a homomorphism of commutative rings. Show that $f$ is a regular epimorphism (in the category of commutative rings) if and only if it is surjective (as a map of sets). In particular, the inclusion map $\mathbf{Z} \hookrightarrow \mathbf{Q}$ of Example 10.3.0.1 is an epimorphism in the category of commutative rings which is not regular.
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