# localization

Let $R$ be a commutative ring and let $S$ be a nonempty multiplicative subset of $R$. The 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 $\mathfrak{p}$ in $R$, the localization of $R$ at $S$ is written $R_{\mathfrak{p}}$.

Title localization Localization 2013-03-22 11:50:21 2013-03-22 11:50:21 djao (24) djao (24) 11 djao (24) Definition msc 13B30 ring of fractions FractionField