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