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.
Search Encyclopedia

Featured Article
