localizationLocalization2001-10-19 13:11:242003-04-02 06:15:51Definition\usepackage{amssymb}
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\usepackage{xypic}Let $R$ be a commutative ring and let $S$ be a nonempty multiplicative subset of $R$. The {\em 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:
\begin{itemize}
\item $(a,s) + (b,t) = (at+bs,st)$
\item $(a,s) \cdot (b,t) = (a \cdot b,s \cdot t)$
\end{itemize}
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 $R_{\mathfrak{p}}$.