Definition:Quotient Epimorphism

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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.