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localization
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(Definition)
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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 $R_{\mathfrak{p}}$ .
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"localization" is owned by djao.
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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 32 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 8607 times total.
Classification:
| AMS MSC: | 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization) |
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Pending Errata and Addenda
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