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localization (Definition)

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 $ \mathfrak{p}$ in $ R$, the localization of $ R$ at $ S$ is written $ R_{\mathfrak{p}}$.



"localization" is owned by djao.
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See Also: fraction field

Other names:  ring of fractions

Attachments:
extension by localization (Definition) by pahio
localization of a module (Definition) by CWoo
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Cross-references: prime ideal, complement, multiplicative set, minimal, multiplication, addition, equivalence relation, equivalence classes, ring, multiplicative subset, commutative ring
There are 23 references to this entry.

This is version 6 of localization, born on 2001-10-19, modified 2003-04-02.
Object id is 391, canonical name is Localization.
Accessed 6824 times total.

Classification:
AMS MSC: 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)
Pending Errata and Addenda
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correction by nerdy2 on 2001-11-08 00:58:26
$R_a$ denotes the localization at the set $\{ a^n \vert n \in Z {\mathrm and } n \geq 0 \}$. The ideal $(a)$ has too many elements.
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non-commutative version by antizeus on 2001-10-19 22:55:05
There's a lovely theory devoted to localization over sets in non-commutative rings that I think deserves some treatment. I'm not up to it at this second though,
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