Field of fractions on arvau

Wikipedia / 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 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.

, and the construction is sometimes also called the fraction field, field of quotients, or quotient field. 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.

Definition

, we define an equivalence relation. We denote the equivalence class, which have the same property with respect to the underlying ring of integers.

Then the field of fractions with addition given by

and multiplication given by

is characterized by the following universal property:

is an injective ring homomorphism.

There is a categorical be the category of integral domains and injective ring maps. The functor 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. Thus the category of fields (which is a full subcategory) is a reflective subcategory.

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.

Examples

The field of fractions of the ring of integers is the field of rationals.
be the ring of Gaussian integers, the field of Gaussian rationals.
The field of fractions of a field is canonically isomorphic to the field itself.
For any field , the field of fractions of the one-variable polynomial ring is the .
For any field , the field of fractions of the formal power series ring is the field of formal Laurent series.
The field of fractions of the convolution ring of half-line functions yields a space of operators, including the Dirac delta function, differential operator, and integral operator. This construction gives an alternate representation of the Laplace transform that does not depend explicitly on an integral transform.

Generalizations

Localization

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

Two special cases of this are notable:

is the complement of a prime ideal. is an integral domain.
is the set of non-zero-divisors 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.

will be the trivial ring.

Semifield of fractions

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 denote the tropical semiring be the polynomial semiring).

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