Field of fractions
|Algebraic structure → Ring theory|
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 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, a related construction is called the localization or ring of quotients.
Given an integral domain and letting , we define an equivalence relation on by letting whenever . We denote the equivalence class of by the fraction . 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 .
- 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 .
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 fractions
with and in .
- Ore condition; condition related to constructing fractions in the noncommutative case.
- Projective line over a ring; alternative structure not limited to integral domains.
- Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189.
- Ėrnest Borisovich Vinberg (2003). A course in algebra. p. 131.
- Stephan Foldes (1994). Fundamental structures of algebra and discrete mathematics. John Wiley & Sons. p. 128.
- Pierre Antoine Grillet (2007). Abstract algebra. p. 124.