# Definition:Quotient

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### 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 $d$.

### 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).

### Quotient Set

- 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 $\map {q_\RR} s = \eqclass s \RR$.

### Quotient Structure

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 $y$.

### Quotient (Topological) Space

Let $T = \struct {S, \tau}$ be a topological space.

Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.

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

Let $\tau_\RR$ be the quotient topology on $S / \RR$ by $q_\RR$:

- $\tau_\RR := \set {U \subseteq S / \RR: q_\RR^{-1} \sqbrk U \in \tau}$

The **quotient space of $S$ by $\RR$** is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$:

- $T_\RR := \struct {S / \RR, \tau_\RR}$

Hence:

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

- The
**quotient topology**on $S / \RR$ by $q_\RR$ is the topology $\tau_{S / \RR}$, also called the identification topology.

### Linear Algebra

## Linguistic Note

The word **quotient** derives from the Latin word meaning **how often**.