Definition:Quotient

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Quotient may refer to:

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

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

Quotient Mapping: The mapping $q_{\mathcal R}: S \to S / \mathcal R$ defined as $q_{\mathcal R} \left({s}\right) = \left[\!\left[{s}\right]\!\right]_{\mathcal R}$.

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

Quotient Structure: If $\mathcal R$ is a congruence for $\circ$ on an algebraic structure $\left({S, \circ}\right)$, and $\circ_{\mathcal R}$ is the operation induced on $S / \mathcal R$ by $\circ$, then $\left({S / \mathcal R, \circ_{\mathcal R}}\right)$ is the quotient structure defined by $\mathcal R$.

Quotient Group: The coset space $G / N$, where $N$ is a normal subgroup of a group $G$, and the group product is defined as $\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$ is called the quotient group of $G$ by $N$.

Quotient Ring: $\left({R / J, +, \circ}\right)$ is the quotient ring of a ring $\left({R, +, \circ}\right)$ and an ideal $J$.

Quotient Field: $\left({F, +, \circ}\right)$ is a quotient field of an integral domain $\left({D, +, \circ}\right)$ iff $\left({F, +, \circ}\right)$ contains $\left({D, +, \circ}\right)$ algebraically such that:

$\displaystyle \forall z \in F: \exists x \in D, y \in D^*: z = \frac x y$

where $\displaystyle \frac x y$ is $x$ divided by $y$.

Let $\left({X, \vartheta}\right)$ be a topological space.

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

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

The Quotient Space is the quotient set $X / \mathcal R$ whose topology $\vartheta_{X / \mathcal R}$ is defined as $U \in \vartheta_{X / \mathcal R} \iff q_\mathcal R^{-1} \left({U}\right) \in \vartheta$.

The Quotient Topology on $X / \mathcal R$ by $q_\mathcal R$ is the topology $\vartheta_{X / \mathcal R}$, also called the identification topology.