Field of fractions

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In abstract algebra, the field of fractions of an integral domain 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 integers and the field of rational numbers. 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 which is not an integral domain, the analogous construction is called the localization or ring of quotients.

DefinitionEdit

Given an integral domain and letting  , we define an equivalence relation on   by letting   whenever  . We denote the equivalence class 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:

if   is an injective 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 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 from the category of fields to  . Thus the category of fields (which is a full subcategory) is a reflective subcategory of  .

A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rng   with no nonzero zero divisors. The embedding is given by   for any nonzero  .[1]

ExamplesEdit

  • 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 to the field itself.
  • Given a field  , the field of fractions of the polynomial ring in one indeterminate   (which is an integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions[2][3][4][5] and is denoted  .

GeneralizationsEdit

LocalizationEdit

For any commutative ring   and any multiplicative set   in  , the localization   is the commutative ring consisting of fractions

 

with   and  , where now   is equivalent to   if and only if there exists   such that  .

Two special cases of this are notable:

  • If   is the complement of a prime ideal  , then   is also denoted  .
    When   is an integral domain and   is the zero ideal,   is the field of fractions of  .
  • If   is the set of non-zero-divisors in  , then   is called the total quotient ring.
    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   to contain 0, but in that case   will be the trivial ring.

Semifield of fractionsEdit

The semifield of fractions of a commutative semiring with no zero divisors is the smallest semifield in which it can be embedded.

The elements of the semifield of fractions of the commutative semiring   are equivalence classes written as

 

with   and   in  .

See alsoEdit

ReferencesEdit

  1. ^ Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189.
  2. ^ Vinberg, Ėrnest Borisovich (2003). A course in algebra. American Mathematical Society. p. 131. ISBN 978-0-8218-8394-5.
  3. ^ Foldes, Stephan (1994). Fundamental structures of algebra and discrete mathematics. Wiley. p. 128. ISBN 0-471-57180-6.
  4. ^ Grillet, Pierre Antoine (2007). "3.5 Rings: Polynomials in One Variable". Abstract algebra. Springer. p. 124. ISBN 978-0-387-71568-1.
  5. ^ Marecek, Lynn; Mathis, Andrea Honeycutt (6 May 2020). Intermediate Algebra 2e. OpenStax. §7.1.