# Definition:Quotient

Jump to navigation
Jump to search

## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Quotient** may refer to:

### Arithmetic

Let $/$ denote the operation of Division on a standard number field $\Q$, $\R$ or $\C$.

Let $q = p / d$.

Then $q$ is the quotient of $p$ (divided) by $q$.

### Algebra

- The quotient of $a$ on division by $b$ is the unique number $q$ such that $a = q b + r, 0 \le r < \size b$ (see the Division Theorem).

### Set theory

- Quotient Set: The set $S / \RR$ of $\RR$-classes of an equivalence relation $\RR$ of a set $S$.

- Quotient Mapping: The mapping $q_\RR: S \to S / \RR$ defined as $\map {q_\RR} s = \eqclass s \RR$.

### Abstract Algebra

The concepts here, although presented in different forms, are all related.

- Quotient Structure: If $\RR$ is a congruence for $\circ$ on an algebraic structure $\struct {S, \circ}$, and $\circ_\RR$ is the operation induced on $S / \RR$ by $\circ$, then $\struct {S / \RR, \circ_\RR}$ is the quotient structure defined by $\RR$.

- Quotient Group: The coset space $G / N$, where $N$ is a normal subgroup of a group $G$, and the group operation is defined as $\paren {a N} \paren {b N} = \paren {a b} N$ is called the quotient group of $G$ by $N$.

- Quotient Ring: $\struct {R / J, +, \circ}$ is the quotient ring of a ring $\struct {R, +, \circ}$ and an ideal $J$.

- Field of Quotients: $\struct {F, +, \circ}$ is a field of quotients of an integral domain $\struct {D, +, \circ}$ if and only if $\struct {F, +, \circ}$ contains $\struct {D, +, \circ}$ algebraically such that:

- $\forall z \in F: \exists x \in D, y \in D^*: z = \dfrac x y$

- where $\dfrac x y$ is $x$ divided by $y$.

### Topology

Let $\struct {X, \tau}$ be a topological space.

Let $\RR \subseteq X^2$ be an equivalence relation on $X$.

Let $q_\RR: X \to X / \RR$ be the quotient mapping induced by $\RR$.

- The Quotient Space is the quotient set $X / \RR$ whose topology $\tau_{X / \RR}$ is defined as $U \in \tau_{X / \RR} \iff q_\RR^{-1} \sqbrk U \in \tau$.

- The Quotient Topology on $X / \RR$ by $q_\RR$ is the topology $\tau_{X / \RR}$, also called the identification topology.