# A Ring Epimorphism that is not a Surjection

## A Ring Epimorphism that is not a Surjection

Consider the inclusion $i : \mathbb{Z} \to \mathbb{Q}$. It is clearly not surjective. However, given any two ring homomorphisms$f ,g : \mathbb{Q} \to R$ such that$f \circ i = g \circ i$, it holds that $f=g$ because ring homomorphisms out of the rationals that agree on the integers must agree everywhere. So, $i$ is an epimorphism in Ring.