# Definition:Quotient Epimorphism

## Definition

Let $\RR$ be a congruence relation on an algebraic structure $\struct {S, \circ}$.

Let $q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}$ denote the quotient mapping from $\struct {S, \circ}$ to the quotient structure $\struct {S / \RR, \circ_\RR}$:

$\forall x \in S: \map {q_\RR} x = \eqclass x \RR$

where $\eqclass x \RR$ denotes the equivalence class of $x$ under $\RR$.

Then $q_\RR$ is referred to as the quotient epimorphism from $\struct {S, \circ}$ to $\struct {S / \RR, \circ_\RR}$.

This is usually encountered in the context of specific algebraic structures thus:

### 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
• a projection.

## Also see

• Results about quotient epimorphisms can be found here.