Given a unital ring (or only a monoid) and a central multiplicative subset (i.e. set containing and with every two elements containing their product, and such that all its elements are central in ) the ring of fractions (monoid of fractions, respectively) is sometimes said to be the commutative localization of at ; the same name is also given to the canonical map of rings (monoids resp.). The ring of fractions is defined as the set of equivalence classes where iff , (if is an integral domain one can skip mentioning in this condition); the equivalence classes are called fractions and denoted ; by centrality of it is easy to guess the multiplication rule and for the addition one first takes the representatives with the same denominator and then adds the numerators. E.g. the formula will do, and we indeed get a ring with unit together with the canonical homomorphism of rings given by .
Localization of commutative rings at multiplicative subsets is the standard example, but the centrality of is enough for the whole theory to pass through.
Commutative localization can be extended to left modules.
Module of fractions is the left -module equipped with the natural map of -modules and defined as follows:
The underlying set of consists of equivalence classes of pairs where iff there exist such that , the multiplication by scalar is defined by and the addition is . The correspondence extends to a functor . The forgetful functor is fully faithful functor and there is a natural transformation of functors whose components are the -module maps given by .
It can be then shown that this elementary approach is equivalent to the definition via the extension of scalars formula .
Commutative localization in which also is commutative is a basic procedure used in defining algebraic scheme as a locally ringed space. Another special case of this procedure is forming the quotient field of a commutative integral domain.