The field of fractions of the ring of is the field of rationals: ${\displaystyle \mathbb {Q} =\operatorname {Frac} (\mathbb {Z} )}$.

integers

Let ${\displaystyle R:=\{a+b\mathrm {i} \mid a,b\in \mathbb {Z} \}}$ be the ring of . Then ${\displaystyle \operatorname {Frac} (R)=\{c+d\mathrm {i} \mid c,d\in \mathbb {Q} \}}$, the field of Gaussian rationals.

Gaussian integers

The field of fractions of a field is canonically to the field itself.

isomorphic

Given a field ${\displaystyle K}$, the field of fractions of the in one indeterminate ${\displaystyle K[X]}$ (which is an integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions^[2][3][4][5] and is denoted ${\displaystyle K(X)}$.

polynomial ring

The field of fractions of the ring of half-line functions yields a space of operators, including the Dirac delta function, differential operator, and integral operator. This construction gives an alternate representation of the Laplace transform that does not depend explicitly on an integral transform.^[6]

convolution

If ${\displaystyle S}$ is the complement of a ${\displaystyle P}$, then ${\displaystyle S^{-1}R}$ is also denoted ${\displaystyle R_{P}}$.
When ${\displaystyle R}$ is an integral domain and ${\displaystyle P}$ is the zero ideal, ${\displaystyle R_{P}}$ is the field of fractions of ${\displaystyle R}$.

prime ideal

If ${\displaystyle S}$ is the set of non- in ${\displaystyle R}$, then ${\displaystyle S^{-1}R}$ is called the total quotient ring.
The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring.

zero-divisors

; condition related to constructing fractions in the noncommutative case.

Ore condition

Total ring of fractions