One can construct the quotient field Quot(R) of the integral domain 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) iff nb=md (we think of the class of (n, d) as the fraction n/d). The embedding is given by n |-> (n,1). 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 quotient field of R is characterized by the following universal property: if f : R -> F is a ring homomorphism from R into a field F, then there exists a unique ring homomorphism g : Quot(R) -> F which extends f.
Assigning to every integral domain its quotient field defines a functor from the category of integral domains to the category of fields. This functor is left adjoint to the forgetful functor which assigns to every field its underlying integral domain.