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.
- 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 and is denoted .
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 .
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 .
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.
- Ore condition; this is the condition one needs to consider in the noncommutative case.
- Projective line over a ring; alternative structure not limited to integral domains.
- Ėrnest Borisovich Vinberg (2003). A course in algebra. p. 131.
- Stephan Foldes (1994). Fundamental structures of algebra and discrete mathematics. p. 128.
- Pierre Antoine Grillet (2007). Abstract algebra. p. 124.
- Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189.