Editing Field of fractions

{{Short description|Abstract algebra concept}}
{{redirect-distinguish|Quotient field|Quotient ring}}
{{Ring theory sidebar}}

In [[abstract algebra]], the '''field of fractions''' of an [[integral domain]] is the smallest [[field (mathematics)|field]] in which it can be [[Embedding|embedded]]. The construction of the field of fractions is modeled on the relationship between the integral domain of [[integer]]s and the field of [[rational number]]s. Intuitively, it consists of ratios between integral domain elements.

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

== Definition ==

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

Then the '''field of fractions''' is the set <math>\text{Frac}(R) = (R \times R^*)/\sim</math> with addition given by
:<math>\frac{n}{d} + \frac{m}{b} = \frac{nb+md}{db}</math>  
and multiplication given by 
:<math>\frac{n}{d} \cdot \frac{m}{b} = \frac{nm}{db}.</math>
One may check that these operations are well-defined and that, for any integral domain <math>R</math>, <math>\text{Frac}(R)</math> is indeed a field. In particular, for <math>n,d \neq 0</math>, the multiplicative inverse of <math>\frac{n}{d}</math> is as expected: <math>\frac{d}{n} \cdot \frac{n}{d} = 1</math>.

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

The field of fractions of <math>R</math> is characterized by the following [[universal property]]:

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

There is a [[category theory|categorical]] interpretation of this construction.  Let <math>\mathbf{C}</math> be the [[category (mathematics)|category]] of integral domains and injective ring maps. The [[functor]] from <math>\mathbf{C}</math> to the [[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 [[adjoint functor|left adjoint]] of the [[inclusion functor]] from the category of fields to <math>\mathbf{C}</math>. Thus the category of fields (which is a full subcategory) is a [[reflective subcategory]] of <math>\mathbf{C}</math>.

A [[multiplicative identity]] is not required for the role of the integral domain; this construction can be applied to any [[zero ring|nonzero]] commutative [[rng (algebra)|rng]] <math>R</math> with no nonzero [[zero divisor]]s. The embedding is given by <math>r\mapsto\frac{rs}{s}</math> for any nonzero <math>s\in R</math>.<ref>{{cite book|last1=Hungerford|first1=Thomas W.|title=Algebra|date=1980|publisher=Springer|location=New York|isbn=3540905189|pages=142–144|edition= Revised 3rd}}</ref>

== Examples ==

* The field of fractions of the ring of [[Integer#Algebraic_properties|integers]] is the field of [[rational number|rationals]]: <math>\Q = \operatorname{Frac}(\Z)</math>.
* Let <math>R:=\{a+b\mathrm{i} \mid a,b \in \Z\}</math> be the ring of [[Gaussian integer]]s. Then <math>\operatorname{Frac}(R)=\{c+d\mathrm{i}\mid c,d\in\Q\}</math>, the field of [[Gaussian rational]]s.
* The field of fractions of a field is canonically [[isomorphism|isomorphic]] to the field itself.
* Given a field <math>K</math>, the field of fractions of the [[polynomial ring]] in one indeterminate <math>K[X]</math> (which is an integral domain), is called the ''{{visible anchor|field of rational functions}}'', ''field of rational fractions'', or ''field of rational expressions''<ref>{{cite book |first=Ėrnest Borisovich  |last=Vinberg |url=https://books.google.com/books?id=rzNq39lvNt0C&pg=PA132 |title=A course in algebra |year=2003 |page=131 |isbn=978-0-8218-8394-5 |publisher=American Mathematical Society}}</ref><ref>{{cite book|first=Stephan |last=Foldes|url=https://archive.org/details/fundamentalstruc0000fold|title=Fundamental structures of algebra and discrete mathematics |publisher=Wiley |year=1994|page=[https://archive.org/details/fundamentalstruc0000fold/page/128 128]|url-access=registration |isbn=0-471-57180-6}}</ref><ref>{{cite book |first=Pierre Antoine |last=Grillet |chapter=3.5 Rings: Polynomials in One Variable |chapter-url=https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA124|title=Abstract algebra|year=2007|page=124 |isbn=978-0-387-71568-1 |publisher=Springer}}</ref><ref>{{cite book|last1 = Marecek | first1 = Lynn | last2 = Mathis | first2 = Andrea Honeycutt | title = Intermediate Algebra 2e | date = 6 May 2020 | publisher = [[OpenStax]] <!-- | location = Houston, Texas -->|  url = https://openstax.org/details/books/intermediate-algebra-2e | at = §7.1}}</ref> and is denoted <math>K(X)</math>.
* The field of fractions of the [[convolution]] ring of half-line functions yields a [[convolution quotient | 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.<ref>{{cite book | first=Jan | last=Mikusiński | url=https://books.google.com/books?id=e8LSBQAAQBAJ | title=Operational Calculus| date=14 July 2014 | publisher=Elsevier | isbn=9781483278933 }}</ref>

== Generalizations ==
=== Localization ===
{{main|Localization (commutative algebra)}}

For any [[commutative ring]] <math>R</math> and any [[multiplicative set]] <math>S</math> in <math>R</math>, the [[localization of a ring|localization]] <math>S^{-1}R</math> is the [[commutative ring]] consisting of [[fraction]]s
:<math>\frac{r}{s}</math>
with <math>r\in R</math> and <math>s\in S</math>, where now <math>(r,s)</math> is equivalent to <math>(r',s')</math> if and only if there exists <math>t\in S</math> such that <math>t(rs'-r's)=0</math>.

Two special cases of this are notable:
* If <math>S</math> is the complement of a [[prime ideal]] <math>P</math>, then <math>S^{-1}R</math> is also denoted <math>R_P</math>.<br/>When <math>R</math> is an [[integral domain]] and <math>P</math> is the zero ideal, <math>R_P</math> is the field of fractions of <math>R</math>.
* If <math>S</math> is the set of non-[[zero-divisor]]s in <math>R</math>, then <math>S^{-1}R</math> is called the [[total quotient ring]].<br/>The [[total quotient ring]] of an [[integral domain]] is its field of fractions, but the [[total quotient ring]] is defined for any [[commutative ring]].

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

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

The elements of the semifield of fractions of the commutative [[semiring]] <math>R</math> are [[equivalence class]]es written as
:<math>\frac{a}{b}</math>
with <math>a</math> and <math>b</math> in <math>R</math> and <math>b\neq 0</math>.

== See also ==
* [[Ore condition]]; condition related to constructing fractions in the noncommutative case.
* [[Total ring of fractions]]

== References ==
{{reflist}}

[[Category:Field (mathematics)]]
[[Category:Commutative algebra]]