For a commutative integral domain , the set of all nonzero elements is a multiplicative set, the corresponding commutative localization is an injective homomorphism of rings and is a field, called the field of fractions or the quotient field of . Its elements are fractions where and which are by the definition the equivalence classes of pairs and iff . The addition is given by the formula
and multiplication by .
The field of fractions is unique minimal field in which the integral domain is embedded in the sense that every field contains the subfield isomorphic to , namely consisting of all the fractions with , taken in the sense of division in .
In the noncommutative case, the notion of a quotient skewfield is not always, and not uniquely defined. One class where this notion works well is the case or (left or right) Ore domains where the set of all nonzero elements form a (left or right) Ore set and the Ore localization at that set defines a skewfield of fractions?.
Not every noncommutative integral domain can be embedded at all into a division ring. Suppose it does, say . Then all the say left fractions with do not form a subfield, namely we can not in general add a fractions to a fraction with common denominator, so this candidate for a subfield may have noninvertible elements.