# Definition:Quotient Epimorphism

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## Contents

## Definition

### Group

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

The mapping $q_N: G \to G / N$ defined as:

- $\forall x \in G: \map {q_N} x = x N$

is known as the **quotient (group) epimorphism** from $G$ to $G / N$.

### Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$.

Let $J$ be an ideal of $R$.

Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$.

The mapping $\phi: R \to R / J$ given by:

- $\forall x \in R: \map \phi x = x + J$

is known as the **quotient (ring) epimorphism from $\struct {R, +, \circ}$ (on)to $\struct {R / J, +, \circ}$**.

## Also known as

A **quotient epimorphism** is also known variously as:

- a
**quotient morphism** - a
**natural epimorphism** - a
**natural morphism** - a
**natural homomorphism** - a
**canonical epimorphism** - a
**canonical morphism**.