localization
Let $R$ be a commutative ring and let $S$ be a nonempty multiplicative subset of $R$. The localization^{} of $R$ at $S$ is the ring ${S}^{1}R$ whose elements are equivalence classes^{} of $R\times S$ under the equivalence relation $(a,s)\sim (b,t)$ if $r(atbs)=0$ for some $r\in S$. Addition^{} and multiplication in ${S}^{1}R$ are defined by:

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$(a,s)+(b,t)=(at+bs,st)$

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$(a,s)\cdot (b,t)=(a\cdot b,s\cdot t)$
The equivalence class of $(a,s)$ in ${S}^{1}R$ is usually denoted $a/s$. For $a\in R$, the localization of $R$ at the minimal^{} multiplicative set containing $a$ is written as ${R}_{a}$. When $S$ is the complement^{} of a prime ideal^{} $\U0001d52d$ in $R$, the localization of $R$ at $S$ is written ${R}_{\U0001d52d}$.
Title  localization 

Canonical name  Localization 
Date of creation  20130322 11:50:21 
Last modified on  20130322 11:50:21 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  11 
Author  djao (24) 
Entry type  Definition 
Classification  msc 13B30 
Synonym  ring of fractions 
Related topic  FractionField 