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