# Definition:Canonical Epimorphism

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It has been suggested that this page or section be merged into Definition:Quotient Epimorphism.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

It has been suggested that this page be renamed.In particular: disambiguation candidateTo discuss this page in more detail, feel free to use the talk page. |

## 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 Quotient Epimorphism is Epimorphism.

This article is complete as far as it goes, but it could do with expansion.In particular: 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.In fact I've found it: Ring Epimorphism from Integers to Integers Modulo m. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

Work In ProgressIn particular: Don't forget to rework the citation chain once the issues on this page have all been addressed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

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