# Definition:Canonical Epimorphism

Jump to navigation
Jump to search

## 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.

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\S 1.2$