localization of a module

Let R be a commutative ring and M an R-module. Let SR be a non-empty multiplicative set. Form the Cartesian productMathworldPlanetmath M×S, and define a binary relationMathworldPlanetmath on M×S as follows:

(m1,s1)(m2,s2) if and only if there is some tS such that t(s2m1-s1m2)=0

Proposition 1.

on M×S is an equivalence relationMathworldPlanetmath.


Clearly (m,s)(m,s) as t(sm-sm)=0 for any tS, where S. Also, (m1,s1)(m2,s2) implies that (m2,s2)(m1,s1), since t(s2m1-s1m2)=0 implies that t(s1m2-s2m1)=0. Finally, given (m1,s1)(m2,s2) and (m2,s2)(m3,s3), we are led to two equations t(s2m1-s1m2)=0 and u(s3m2-s2m3)=0 for some t,uS. Expanding and rearranging these, then multiplying the first equation by us3 and the second by ts1, we get tus2(s3m1-s1m3)=0. Since tus2S, (m1,s1)(m3,s3) as required. ∎

Let MS be the set of equivalence classesMathworldPlanetmath in M×S under . For each (m,s)M×S, write

[(m,s)] or more commonly ms

the equivalence class in MS containing (m,s). Next,

  • define a binary operationMathworldPlanetmath + on MS as follows:

  • define a function :RS×MSMS as follows:


    where RS is the localizationMathworldPlanetmath of R over S.

Proposition 2.

MS together with + and defined above is a unital module over RS.


That + and are well-defined is based on the following: if (m1,s1)(m2,s2), then


which are clear by PropositionPlanetmathPlanetmath 1. Furthermore + is commutativePlanetmathPlanetmathPlanetmathPlanetmath and associative and that distributes over + on both sides, which are all properties inherited from M. Next, 0s is the additive identity in MS and -msMS is the additive inverse of ms. So MS is a module over RS. Finally, since (mt,st)(m,s) for any tS, ttms=ms so that MS is unital. ∎

Definition. MS, as an RS-module, is called the localization of M at S. MS is also written S-1M.


  • The notion of the localization of a module generalizes that of a ring in the sense that RS is the localization of R at S as an RS-module.

  • If S=R-𝔭, where 𝔭 is a prime idealMathworldPlanetmathPlanetmathPlanetmath in R, then MS is usually written M𝔭.

Title localization of a module
Canonical name LocalizationOfAModule
Date of creation 2013-03-22 17:26:59
Last modified on 2013-03-22 17:26:59
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 13B30