# Field of fractions

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"Quotient field" redirects here. Not to be confused with [Quotient ring](/source/Quotient_ring).

In [abstract algebra](/source/Abstract_algebra), the **field of fractions** of an [integral domain](/source/Integral_domain) is the smallest [field](/source/Field_(mathematics)) in which it can be [embedded](/source/Embedding). The construction of the field of fractions is modeled on the relationship between the integral domain of [integers](/source/Integer) and the field of [rational numbers](/source/Rational_number). Intuitively, it consists of ratios between integral domain elements.

The field of fractions of an integral domain R is sometimes denoted by \operatorname{Frac}(R) or \operatorname{Quot}(R), and the construction is sometimes also called the **fraction field**, **field of quotients**, or **quotient field** of R. All four are in common usage, but are not to be confused with the [quotient of a ring by an ideal](/source/Quotient_ring), which is a quite different concept. For a [commutative ring](/source/Commutative_ring) that is not an integral domain, the analogous construction is called the [localization](/source/Localization_(commutative_algebra)) or ring of quotients.

## Definition

Given an integral domain R and letting R^* = R \setminus \{0\}, we define an [equivalence relation](/source/Equivalence_relation) on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the [equivalence class](/source/Equivalence_class) of (n,d) by \frac{n}{d}. This notion of equivalence is motivated by the rational numbers \Q, which have the same property with respect to the underlying [ring](/source/Ring_(mathematics)) \Z of integers.

Then the **field of fractions** is the set \text{Frac}(R) = (R \times R^*)/\sim with addition given by

- \frac{n}{d} + \frac{m}{b} = \frac{nb+md}{db}

and multiplication given by

- \frac{n}{d} \cdot \frac{m}{b} = \frac{nm}{db}.

One may check that these operations are well-defined and that, for any integral domain R, \text{Frac}(R) is indeed a field. In particular, for n,d \neq 0, the multiplicative inverse of \frac{n}{d} is as expected: \frac{d}{n} \cdot \frac{n}{d} = 1.

The embedding of R in \operatorname{Frac}(R) maps each n in R to the fraction \frac{en}{e} for any nonzero e\in R (the equivalence class is independent of the choice e). This is modeled on the identity \frac{n}{1}=n.

The field of fractions of R is characterized by the following [universal property](/source/Universal_property):

- if h: R \to F is an [injective](/source/Injective) [ring homomorphism](/source/Ring_homomorphism) from R into a field F, then there exists a unique ring homomorphism g: \operatorname{Frac}(R) \to F that extends h.

There is a [categorical](/source/Category_theory) interpretation of this construction. Let \mathbf{C} be the [category](/source/Category_(mathematics)) of integral domains and injective ring maps. The [functor](/source/Functor) from \mathbf{C} to the [category of fields](/source/Category_of_fields) that takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the [left adjoint](/source/Adjoint_functor) of the [inclusion functor](/source/Inclusion_functor) from the category of fields to \mathbf{C}. Thus the category of fields (which is a full subcategory) is a [reflective subcategory](/source/Reflective_subcategory) of \mathbf{C}.

A [multiplicative identity](/source/Multiplicative_identity) is not required for the role of the integral domain; this construction can be applied to any [nonzero](/source/Zero_ring) commutative [rng](/source/Rng_(algebra)) R with no nonzero [zero divisors](/source/Zero_divisor). The embedding is given by r\mapsto\frac{rs}{s} for any nonzero s\in R.[1]

## Examples

- The field of fractions of the ring of [integers](/source/Integer#Algebraic_properties) is the field of [rationals](/source/Rational_number): \Q = \operatorname{Frac}(\Z).
- Let R:=\{a+b\mathrm{i} \mid a,b \in \Z\} be the ring of [Gaussian integers](/source/Gaussian_integer). Then \operatorname{Frac}(R)=\{c+d\mathrm{i}\mid c,d\in\Q\}, the field of [Gaussian rationals](/source/Gaussian_rational).
- The field of fractions of a field is canonically [isomorphic](/source/Isomorphism) to the field itself.
- For any field k, the field of fractions of the one-variable [polynomial ring](/source/Polynomial_ring) k[t] is the k(t).[2][3][4][5]
- For any field k, the field of fractions of the [formal power series ring](/source/Formal_power_series_ring) k[\![t]\!] is the field of [formal Laurent series](/source/Formal_Laurent_series) k(\!(t)\!).
- The field of fractions of the [convolution](/source/Convolution) ring of half-line functions yields a [space of operators](/source/Convolution_quotient), including the [Dirac delta function](/source/Dirac_delta_function), [differential operator](/source/Differential_operator), and [integral operator](/source/Integral_operator). This construction gives an alternate representation of the [Laplace transform](/source/Laplace_transform) that does not depend explicitly on an integral transform.[6]

## Generalizations

### Localization

Main article: [Localization (commutative algebra)](/source/Localization_(commutative_algebra))

For any [commutative ring](/source/Commutative_ring) R and any [multiplicative set](/source/Multiplicative_set) S in R, the [localization](/source/Localization_of_a_ring) S^{-1}R is the [commutative ring](/source/Commutative_ring) consisting of [fractions](/source/Fraction)

- \frac{r}{s}

with r\in R and s\in S, where now (r,s) is equivalent to (r',s') if and only if there exists t\in S such that t(rs'-r's)=0.

Two special cases of this are notable:

- If S is the complement of a [prime ideal](/source/Prime_ideal) P, then S^{-1}R is also denoted R_P. When R is an [integral domain](/source/Integral_domain) and P is the zero ideal, R_P is the field of fractions of R.
- If S is the set of non-[zero-divisors](/source/Zero-divisor) in R, then S^{-1}R is called the [total quotient ring](/source/Total_quotient_ring). The [total quotient ring](/source/Total_quotient_ring) of an [integral domain](/source/Integral_domain) is its field of fractions, but the [total quotient ring](/source/Total_quotient_ring) is defined for any [commutative ring](/source/Commutative_ring).

Note that it is permitted for S to contain 0, but in that case S^{-1}R will be the [trivial ring](/source/Trivial_ring).

### Semifield of fractions

The **semifield of fractions** of a [commutative semiring](/source/Commutative_semiring) in which every nonzero element is (multiplicatively) cancellative is the smallest [semifield](/source/Semifield) in which it can be [embedded](/source/Embedding). (Note that, unlike the case of rings, a semiring with no [zero divisors](/source/Zero_divisor) can still have nonzero elements that are not cancellative. For example, let \mathbb{T} denote the [tropical semiring](/source/Tropical_semiring) and let R=\mathbb{T}[X] be the [polynomial semiring](/source/Polynomial_ring#polynomial_rigs) over \mathbb{T}. Then R has no zero divisors, but the element 1+X is not cancellative because (1+X)(1+X+X^2)=1+X+X^2+X^3=(1+X)(1+X^2)).

The elements of the semifield of fractions of the commutative [semiring](/source/Semiring) R are [equivalence classes](/source/Equivalence_class) written as

- \frac{a}{b}

with a and b in R and b\neq 0.

## See also

- [Ore condition](/source/Ore_condition); condition related to constructing fractions in the noncommutative case.
- [Total ring of fractions](/source/Total_ring_of_fractions)

## References

1. Hungerford, Thomas W. (1980). *Algebra*. Revised 3rd ed. New York: Springer. pp. 142–144. ISBN 3540905189.

1. Vinberg, Ėrnest Borisovich (2003). [*A course in algebra*](https://books.google.com/books?id=rzNq39lvNt0C&pg=PA132). American Mathematical Society. p. 131. ISBN 978-0-8218-8394-5.

1. Foldes, Stephan (1994). [*Fundamental structures of algebra and discrete mathematics*](https://archive.org/details/fundamentalstruc0000fold). Wiley. p. [128](https://archive.org/details/fundamentalstruc0000fold/page/128). ISBN 0-471-57180-6.

1. Grillet, Pierre Antoine (2007). ["3.5 Rings: Polynomials in One Variable"](https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA124). *Abstract algebra*. Springer. p. 124. ISBN 978-0-387-71568-1.

1. Marecek, Lynn & Mathis, Andrea Honeycutt (6 May 2020). [*Intermediate Algebra 2e*](https://openstax.org/details/books/intermediate-algebra-2e). [OpenStax](/source/OpenStax). §7.1.

1. Mikusiński, Jan (14 July 2014). [*Operational Calculus*](https://books.google.com/books?id=e8LSBQAAQBAJ). Elsevier. ISBN 9781483278933.

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Adapted from the Wikipedia article [Field of fractions](https://en.wikipedia.org/wiki/Field_of_fractions) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Field_of_fractions?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.