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 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 Progress In 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): $\S 1.2$