|
|
|
Revision difference : localization |
| Version current |
Version 5 |
|
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:
|
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 $A \times S$ 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} |
\begin{itemize} |
| \item $(a,s) + (b,t) = (at+bs,st)$ |
\item $(a,s) + (b,t) = (at+bs,st)$ |
| \item $(a,s) \cdot (b,t) = (a \cdot b,s \cdot t)$ |
\item $(a,s) \cdot (b,t) = (a \cdot b,s \cdot t)$ |
| \end{itemize} |
\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 $R_{\mathfrak{p}}$. |
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 $R_{\mathfrak{p}}$. |
|
|
|
|
