\begin{definition}[Definition:Canonical Epimorphism]
Let $m \in \Z$.
Let $f:\Z \to \Z_m$ be a mapping such that:
:$\forall n \in \Z: \map f n = \eqclass n m$
where:
::$\Z_m$ denotes the integers modulo $m$.
::$\eqclass n m$ denotes the residue class of $n$ modulo $m$.
Then $f$ is referred to as the '''canonical epimorphism''' ( '''from $\Z$ to $\Z_m$''').
That this is an epimorphism is proved in Quotient Epimorphism is Epimorphism.
{{expand|Note that this definition is unnecessarily specific - it is an instance of the concept as applied to the integers modulo $m$. It may be appropriate to link to the fuller definition as given in Definition:Quotient Epimorphism, or depending on the context in Hungerford it may be better to rewrite this as a proof that it is such a specific instance. OTOH that ought already to have been documented somewhere (search around) as this area of group theory has already been covered in considerable detail.<br>In fact I've found it: Ring Epimorphism from Integers to Integers Modulo m.}}
\end{definition}