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 $r(at-bs)=0$ 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 $𝔭$ in $R$, the localization of $R$ at $S$ is written $R_{𝔭}$.

Title	localization
Canonical name	Localization
Date of creation	2013-03-22 11:50:21
Last modified on	2013-03-22 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