Field of fractions
Encyclopedia
In abstract algebra
, the field of fractions or field of quotients of an integral domain is the smallest field
in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0. The field of fractions of R is sometimes denoted by Quot(R) or Frac(R).
Mathematicians refer to this construction as the quotient field, field of fractions, or fraction field. All three are in common usage, and which is used is a matter of personal taste. The expression "quotient field" may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
A multiplicative identity is not required; this construction can be applied to any non-trivial commutative
pseudo-ring with no zero divisor
s.
s and at least one nonzero element e. One can construct the field of fractions Quot(R) of R as follows: Quot(R) is the set of equivalence classes
of pairs (n, d), where n and d are elements of R and d is not 0, and the equivalence relation
is:
(n, d) is equivalent to (m, b) if and only if
nb=md (we think of the class of (n, d) as the fraction n/d).
The sum of the equivalence classes of (n, d) and (m, b) is the class of (nb + md, db) and their product is the class of (mn, db). The embedding is given by mapping n to the equivalence class of (en, e). Note that this embedding does not depend on the choice of e. If additionally, R contains a multiplicative identity (that is, R is an integral domain), (en, e) will be equivalent to (n, 1).
The field of fractions of R is characterized by the following universal property
: if f : R → F is an injective ring homomorphism
from R into a field F, then there exists a unique ring homomorphism g : Quot(R) → F which extends f.
There is a categorical
interpretation of this construction. Let C be the category of integral domains and injective ring maps. The functor
from C 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 forgetful functor
from the category of fields to C.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the field of fractions or field of quotients of an integral domain is the smallest field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0. The field of fractions of R is sometimes denoted by Quot(R) or Frac(R).
Mathematicians refer to this construction as the quotient field, field of fractions, or fraction field. All three are in common usage, and which is used is a matter of personal taste. The expression "quotient field" may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
A multiplicative identity is not required; this construction can be applied to any non-trivial commutative
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
pseudo-ring with no zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s.
Examples
- The field of fractions of the ring of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is the field of rationalsRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, Q = Quot(Z). - Let R := { a + b i | a,b in Z } be the ring of Gaussian integerGaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s. Then Quot(R) = {c + d i | c,d in Q}, the field of Gaussian rationalGaussian rationalIn mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers....
s. - The field of fractions of a field is isomorphicIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the field itself. - Given a field K, the field of fractions of the polynomial ringPolynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
in one indeterminate K[X] (which is an integral domain), is called the or field of rational fractions and is denoted K(X).
Construction
Let R be any commutative pseudo-ring without zero divisorZero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s and at least one nonzero element e. One can construct the field of fractions Quot(R) of R as follows: Quot(R) is the set of equivalence classes
of pairs (n, d), where n and d are elements of R and d is not 0, and the equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
is:
(n, d) is equivalent to (m, b) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
nb=md (we think of the class of (n, d) as the fraction n/d).
The sum of the equivalence classes of (n, d) and (m, b) is the class of (nb + md, db) and their product is the class of (mn, db). The embedding is given by mapping n to the equivalence class of (en, e). Note that this embedding does not depend on the choice of e. If additionally, R contains a multiplicative identity (that is, R is an integral domain), (en, e) will be equivalent to (n, 1).
The field of fractions of R is characterized by the following universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
: if f : R → F is an injective ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
from R into a field F, then there exists a unique ring homomorphism g : Quot(R) → F which extends f.
There is a categorical
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
interpretation of this construction. Let C be the category of integral domains and injective ring maps. The functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from C 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 forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
from the category of fields to C.
See also
- Total ring of fractions — a generalization of the field of fractions to rings with zero divisors.
- Localization of a ringLocalization of a ringIn abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
— which generalizes both the field of fractions and the total ring of fractions. - Quotient ringQuotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
— although the quotient rings may be fields, they are entirely distinct from the field of quotients.