In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, the field of fractions of an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
is the smallest
field in which it can be
embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. Intuitively, it consists of ratios between integral domain elements.
The field of fractions of
is sometimes denoted by
or
, and the construction is sometimes also called the fraction field, field of quotients, or quotient field of
. All four are in common usage, but are not to be confused with the
quotient of a ring by an ideal, which is a quite different concept. For a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
which is not an integral domain, the analogous construction is called the
localization or ring of quotients.
Definition
Given an integral domain and letting
, we define an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on
by letting
whenever
. We denote the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of
by
. This notion of equivalence is motivated by the rational numbers
, which have the same property with respect to the underlying
ring of integers.
Then the field of fractions is the set
with addition given by
:
and multiplication given by
:
One may check that these operations are well-defined and that, for any integral domain
,
is indeed a field. In particular, for
, the multiplicative inverse of
is as expected:
.
The embedding of
in
maps each
in
to the fraction
for any nonzero
(the equivalence class is independent of the choice
). This is modeled on the identity
.
The field of fractions of
is characterized by the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
:
:if
is an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
ring homomorphism from
into a field
, then there exists a unique ring homomorphism
which extends
.
There is a
categorical interpretation of this construction. Let
be the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of integral domains and injective ring maps. The
functor from
to the
category of fields which 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 of the
inclusion functor
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
from the category of fields to
. Thus the category of fields (which is a full subcategory) is a
reflective subcategory
In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A' ...
of
.
A
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
is not required for the role of the integral domain; this construction can be applied to any
nonzero commutative
rng with no nonzero
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s. The embedding is given by
for any nonzero
.
Examples
* The field of fractions of the ring of
integers is the field of
rationals:
.
* Let
be the ring of
Gaussian integers. Then
, the field of
Gaussian rationals.
* The field of fractions of a field is canonically
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the field itself.
* Given a field
, the field of fractions of the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in one indeterminate