# 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

- Quotient Mapping on Structure is Epimorphism, where it is shown that $q_\RR$ is indeed an epimorphism.

- Results about
**quotient epimorphisms**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms