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==Localization of a module ==
Let {{mvar|<math>R}}</math> be a [[commutative ring]], {{mvar|<math>S}}</math> be a [[multiplicative set]] in {{mvar|<math>R}}</math>, and {{mvar|<math>M}}</math> be an {{mvar|<math>R}}</math>-[[module (mathematics)|module]]. The '''localization of the module''' {{mvar|<math>M}}</math> by {{mvar|<math>S}}</math>, denoted {{<math|''>S''<sup>−1^{-1}M</supmath>''M''}}, is an {{<math|''>S''<sup>−1^{-1}R</supmath>''R''}}-module that is constructed exactly as the localization of {{mvar|<math>R}}</math>, except that the numerators of the fractions belong to {{mvar|<math>M}}</math>. That is, as a set, it consists of [[equivalence class]]es, denoted <math>\frac ms</math>, of pairs {{<math|>(''m'', ''s'')}}</math>, where <math>m\in M</math> and <math>s\in S,</math> and two pairs {{<math|>(''m'', ''s'')}}</math> and {{<math|>(''n'', ''t'')}}</math> are equivalent if there is an element {{mvar|<math>u}}</math> in {{mvar|<math>S}} </math> such that
:<math>u(sn-tm)=0.</math>
Addition and scalar multiplication are defined as for usual fractions (in the following formula, <math>r\in R,</math> <math>s,t\in S,</math> and <math>m,n\in M</math>):
:<math>\frac{m}{s} + \frac{n}{t} = \frac{tm+sn}{st},</math>
:<math>\frac rs \frac{m}{t} = \frac{r m}{st}.</math>
Moreover, {{<math|''>S''<sup>−1^{-1}M</supmath>''M''}} is also an {{mvar|<math>R}}</math>-module with scalar multiplication
:<math> r\, \frac{m}{s} = \frac r1 \frac ms = \frac{rm}s.</math>
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