Field of fractions

"Quotient field" redirects here. It should not be confused with a quotient ring.

In mathematics, the field of fractions or field of quotients of a zero-divisor free commutative ring with more than one element is the smallest field in which it can be embedded.^[1] It is common to define the field of fractions only for an integral domain, but in fact it exists if and only if the ring has more than one element, is commutative, and has no zero divisors. The elements of the field of fractions of the ring R have the form a/b with a and b in R and b ≠ 0. The field of fractions of the ring R is sometimes denoted by Quot(R) or Frac(R).

Mathematicians refer to this construction as the quotient field, field of fractions, or fraction field. All three are in common usage, and which is used is a matter of personal taste. Those who favor the latter two sometimes claim that the name quotient field incorrectly suggests that the construction is related to taking a quotient of the ring by an ideal.

Examples

• The field of fractions of the ring of integers is the field of rationals, Q = Quot(Z).
• Let R := { a + b i | a,b in Z } be the ring of Gaussian integers. Then Quot(R) = {c + d i | c,d in Q}, the field of Gaussian rationals.
• The field of fractions of a field is isomorphic to the field itself.
• Given a field K, the field of fractions of the polynomial ring in one indeterminate K[X] (which is an integral domain), is called field of rational functions and denoted K(X).

Construction

Let R be any commutative ring without zero divisors and at least one nonzero element e. One can construct the field of fractions Quot(R) of R as follows: Quot(R) is the set of equivalence classes of pairs (n, d), where n and d are elements of R and d is not 0, and the equivalence relation is: (n, d) is equivalent to (m, b) if and only if nb=md (we think of the class of (n, d) as the fraction n/d). The sum of the equivalence classes of (n, d) and (m, b) is the class of (nb + md, db) and their product is the class of (mn, db). The embedding is given by mapping n to the equivalence class of (en, e). Note that this embedding does not depend on the choice of e. If additionally, R contains a multiplicative identity (that is, R is an integral domain), (en, e) will be equivalent to (n, 1).

The field of fractions of R is characterized by the following universal property: if f : R → F is an injective ring homomorphism from R into a field F, then there exists a unique ring homomorphism g : Quot(R) → F which extends f.

There is a categorical interpretation of this construction. Let C be the category of integral domains and injective ring maps. The functor from C 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 C.

References

See also

• Total ring of fractions — a generalization of the field of fractions to rings with zero divisors.
• Localization of a ring — which generalizes both the field of fractions and the total ring of fractions.
• Quotient ring — although the quotient rings may be fields, they are entirely distinct from the field of quotients.