Example 10.3.0.1. Let $\mathbf{Q}$ denote the field of rational numbers, let $\mathbf{Z} \subseteq \mathbf{Q}$ denote the ring of integers, and let $f: \mathbf{Z} \hookrightarrow \mathbf{Q}$ denote the inclusion map. Then $f$ is both a monomorphism and an epimorphism in the category of commutative rings. Consequently, the ring homomorphism $f$ admits (at least) two factorizations as an epimorphism followed by a monomorphism, given by the diagrams
\[ \mathbf{Z} \xrightarrow {\operatorname{id}} \mathbf{Z} \xrightarrow {f} \mathbf{Q} \quad \quad \mathbf{Z} \xrightarrow {f} \mathbf{Q} \xrightarrow {\operatorname{id}} \mathbf{Q}. \]