Field of Fractions - Construction

Construction

Let R be any commutative pseudo-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, d ∈ R and d ≠ 0, such that (n, d) is equivalent to (m, b) if and only if nb=md. This generalizes the property from the rational numbers that n/d=m/b iff nb=md. 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 pairs (n, d) from Quot(R) are usually written .

The embedding is given by mapping n to an equivalence class (en, e). This generalizes the identity n/1=n. 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 h : 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 h.

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.