You are here
Homelocalization
Primary tabs
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:

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

$(a,s)\cdot(b,t)=(a\cdot b,s\cdot t)$
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 $\mathfrak{p}$ in $R$, the localization of $R$ at $S$ is written $R_{{\mathfrak{p}}}$.
Related:
FractionField
Synonym:
ring of fractions
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
13B30 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz
Mar 26
new correction: Misspelled name by DavidSteinsaltz
Mar 21
new correction: underlinetypo by Filipe
Mar 19
new correction: cocycle pro cocyle by pahio
Mar 7
new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier
new image: expected waiting time by robert_dodier
new image: plot W(t) = P(waiting time <= t) by robert_dodier
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz
Mar 26
new correction: Misspelled name by DavidSteinsaltz
Mar 21
new correction: underlinetypo by Filipe
Mar 19
new correction: cocycle pro cocyle by pahio
Mar 7
new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier
new image: expected waiting time by robert_dodier
new image: plot W(t) = P(waiting time <= t) by robert_dodier
Comments
noncommutative version
There's a lovely theory devoted to localization over sets in noncommutative rings that I think deserves some treatment. I'm not up to it at this second though,
correction
$R_a$ denotes the localization at the set $\{ a^n \vert n \in Z {\mathrm and } n \geq 0 \}$. The ideal $(a)$ has too many elements.
Re: correction
oops, i found the 'correct' thing... i'm a dumbass :)