# Definition:Canonical Epimorphism

## Definition

Let $m \in \Z$.

Let $f:\Z \to \Z_m$ be a mapping such that:

- $\forall n \in \Z: f(n) = \left[\!\left[{n}\right]\!\right]_m$

where:

- $\Z_m$ denotes the integers modulo $m$.

- $\left[\!\left[{n}\right]\!\right]_m$ 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.

## Sources

- 1974: Thomas W. Hungerford:
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