Definition:Canonical Epimorphism
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Definition
Let $m \in \Z$.
Let $f:\Z \to \Z_m$ be a mapping such that:
- $\forall n \in \Z: f(n) = \left[\!\left[{n}\right]\!\right]_m$
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.
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.
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Sources
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- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
