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commutative localization

Given a unital ring (or only a monoid) $R$ and a central multiplicative subset $S\subset Z(R)\subset R$ (i.e. set containing $1\in R$ and with every two elements containing their product, and such that all its elements are central in $R$) the ring of fractions (monoid of fractions, respectively) $S^{-1}R$ is sometimes said to be the commutative localization of $R$ at $S$; the same name is also given to the canonical map $R\to S^{-1}R$ of rings (monoids resp.). The ring of fractions is defined as the set of equivalence classes $(s,r)\in S\times R$ where $(s,r)\sim (t,r\prime )$ iff $\exists u\in S$, $usr\prime =utr$ (if $R$ is an integral domain one can skip mentioning $u$ in this condition); the equivalence classes are called fractions and denoted $s^{-1}r$; by centrality of $S$ it is easy to guess the multiplication rule $s^{-1}r{t}^{-1}r\prime =(ts{)}^{-1}(rr\prime )$ and for the addition one first takes the representatives with the same denominator and then adds the numerators. E.g. the formula $s^{-1}r+t^{-1}r\prime =(ts{)}^{-1}(tr+sr\prime )$ will do, and we indeed get a ring $S^{-1}R$ with unit $1^{-1}1$ together with the canonical homomorphism of rings $R\to S^{-1}R$ given by $r\mapsto 1^{-1}r$.

Localization of commutative rings at multiplicative subsets is the standard example, but the centrality of $S$ is enough for the whole theory to pass through.

Commutative localization can be extended to left modules.

Module of fractions $S^{-1}M$ is the left $S^{-1}R$-module $S^{-1}M$ equipped with the natural map of $R$-modules $M\to S^{-1}M$ and defined as follows:

The underlying set of $S^{-1}M$ consists of equivalence classes $s^{-1}m$ of pairs $(s,m)\in S\times M$ where $(s,m)\cong (t,n)$ iff there exist $u\in S$ such that $utm=usn$, the multiplication by scalar is defined by $(s^{-1}r)(t^{-1}m):=(ts{)}^{-1}(rm)$ and the addition is $s^{-1}m+t^{-1}n:=(st{)}^{-1}(tm+sn)$. The correspondence $Q_S:M\mapsto S^{-1}M$ extends to a functor ${}_R\mathrm{Mod}\to {}_{S^{-1}R}\mathrm{Mod}$. The forgetful functor $U:{}_{S^{-1}R}\mathrm{Mod}\to {}_R\mathrm{Mod}$ is fully faithful functor and there is a natural transformation of functors $\mathrm{Id}\to U{Q}_S$ whose components are the $R$-module maps $M\to S^{-1}M$ given by $m\mapsto 1^{-1}m$.

It can be then shown that this elementary approach is equivalent to the definition via the extension of scalars formula $S^{-1}M=S^{-1}R{\otimes }_{R}M$.

The basic result is that the commutative localization $S^{-1}R$ is a flat left module over $R$, the property which holds for more general Ore localization.

Commutative localization in which also $R$ 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.