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Homelocalization
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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
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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 :)