Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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 an integral domain
, 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 that is not an integral domain, the analogous construction is called the localization or ring of quotients.
Given an integral domain
, we define an equivalence relation on
We denote the equivalence class of
This notion of equivalence is motivated by the rational numbers
, which have the same property with respect to the underlying ring
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
, the multiplicative inverse of
(the equivalence class is independent of the choice
is characterized by the following universal property: There is a categorical interpretation of this construction.
be the category of integral domains and injective ring maps.
to the category of fields that 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.
[1] For any commutative ring
is the commutative ring consisting of fractions with
Two special cases of this are notable: Note that it is permitted for
The semifield of fractions of a commutative semiring in which every nonzero element is (multiplicatively) cancellative is the smallest semifield in which it can be embedded.
(Note that, unlike the case of rings, a semiring with no zero divisors can still have nonzero elements that are not cancellative.
denote the tropical semiring and let
The elements of the semifield of fractions of the commutative semiring
are equivalence classes written as with