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Field of fractions

In abstract algebra, the field of fractions 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 are equivalence classes (see the construction below) written as with and in and . The field of fractions of is sometimes denoted by or .

Mathematicians refer to this construction as the field of fractions, fraction field, field of quotients, or quotient field. All four are in common usage. 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.

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 or field of rational fractions[1][2][3] and is denoted  .

ConstructionEdit

Let   be any integral domain. For   with  , the fraction   denotes the equivalence class of pairs  , where   is equivalent to   if and only if  . (The definition of equivalence is modelled on the property of rational numbers that   if and only if  .) The field of fractions   is defined as the set of all such fractions  . The sum of   and   is defined as  , and the product of   and   is defined as   (one checks that these are well defined).

The embedding of   in   maps each   in   to the fraction   for any nonzero   (the equivalence class is independent of the choice  ). This is modelled on the identity  .

The field of fractions of   is characterised 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 forgetful functor from the category of fields to  .

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  .[4]

GeneralizationEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ Ėrnest Borisovich Vinberg (2003). A course in algebra. p. 131.
  2. ^ Stephan Foldes (1994). Fundamental structures of algebra and discrete mathematics. p. 128.
  3. ^ Pierre Antoine Grillet (2007). Abstract algebra. p. 124.
  4. ^ Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189.