field of fractions

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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 te ...
, 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 se ...
is the smallest
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 ratio ...
s. Intuitively, it consists of ratios between integral domain elements. The field of fractions of $R$ is sometimes denoted by $\operatorname\left(R\right)$ or $\operatorname\left(R\right)$, 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, 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 $R^* = R \setminus \$, 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 $R \times R^*$ by letting $\left(n,d\right) \sim \left(m,b\right)$ whenever $nb = md$. 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 $\left(n,d\right)$ by $\frac$. This notion of equivalence is motivated by the rational numbers $\Q$, which have the same property with respect to the underlying ring $\Z$ of integers. Then the field of fractions is the set $\text\left(R\right) = \left(R \times R^*\right)/\sim$ with addition given by :$\frac + \frac = \frac$ and multiplication given by :$\frac \cdot \frac = \frac.$ One may check that these operations are well-defined and that, for any integral domain $R$, $\text\left(R\right)$ is indeed a field. In particular, for $n,d \neq 0$, the multiplicative inverse of $\frac$ is as expected: $\frac \cdot \frac = 1$. The embedding of $R$ in $\operatorname\left(R\right)$ maps each $n$ in $R$ to the fraction $\frac$ for any nonzero $e\in R$ (the equivalence class is independent of the choice $e$). This is modeled on the identity $\frac=n$. The field of fractions of $R$ is characterized by the following universal property: :if $h: R \to F$ 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 In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
from $R$ into a field $F$, then there exists a unique ring homomorphism $g: \operatorname\left(R\right) \to F$ which extends $h$. There is a categorical interpretation of this construction. Let $\mathbf$ 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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
from $\mathbf$ to the
category of fields In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is ...
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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
of the inclusion functor from the category of fields to $\mathbf$. 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 $\mathbf$. 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 s ...
is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rng $R$ 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 zero ...
s. The embedding is given by $r\mapsto\frac$ for any nonzero $s\in R$.

# Examples

* The field of fractions of the ring of
integers 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 ...
is the field of
rationals 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 rati ...
: $\Q = \operatorname\left(\Z\right)$. * Let $R:=\$ be the ring of
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s. Then $\operatorname\left(R\right)=\$, the field of
Gaussian rational In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained b ...
s. * 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 $K$, 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 variab ...
in one indeterminate

# Generalizations

## Localization

For any
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 ...
$R$ and any multiplicative set $S$ in $R$, the localization $S^R$ is the
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 ...
consisting of
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s :$\frac$ with $r\in R$ and $s\in S$, where now $\left(r,s\right)$ is equivalent to $\left(r\text{'},s\text{'}\right)$ if and only if there exists $t\in S$ such that $t\left(rs\text{'}-r\text{'}s\right)=0$. Two special cases of this are notable: * If $S$ is the complement of a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
$P$, then $S^R$ is also denoted $R_P$.
When $R$ is 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 se ...
and $P$ is the zero ideal, $R_P$ is the field of fractions of $R$. * If $S$ is the set of non-
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 zero ...
s in $R$, then $S^R$ is called the
total quotient ring In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
.
The
total quotient ring In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
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 se ...
is its field of fractions, but the
total quotient ring In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
is defined for any
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 ...
. Note that it is permitted for $S$ to contain 0, but in that case $S^R$ will be the
trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
.

## Semifield of fractions

The semifield of fractions of a
commutative semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
with no
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 zero ...
s is the smallest
semifield In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Overview The term semifield has two conflicting meanings, both of which inc ...
in which it can be embedded. The elements of the semifield of fractions of the commutative
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
$R$ are
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 ...
es written as :$\frac$ with $a$ and $b$ in $R$.