# Definition:Canonical Epimorphism

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## Definition

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 Canonical Epimorphism is Epimorphism.