PlanetMath: extension by localization

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extension by localization

(Definition)

Let be a commutative ring and let be a non-empty multiplicative subset of . Then the localisation of at gives the commutative ring $S^{-1}R$ but, generally, it has no subring isomorphic to . Formally, $S^{-1}R$ consists of all elements $\frac{a}{s}$ ( $a \in R$ , $s \in S$ ). Therefore, $S^{-1}R$ is called also a ring of quotients of . If $0 \in S$ , then $S^{-1}R = \{0\}$ ; we assume now that $0 \notin S$ .

The mapping $a \mapsto \frac{as}{s}$ , where is any element of , is well-defined and a homomorphism from to $S^{-1}R$ . All elements of are mapped to units of $S^{-1}R$ .
If, especially, contains no zero divisors of the ring , then the above mapping is an isomorphism from to a certain subring of $S^{-1}R$ , and we may think that $S^{-1}R \supseteq R$ . In this case, the ring of fractions of is an extension ring of ; this concerns of course the case that is an integral domain. But if is a finite ring, then $S^{-1}R = R$ , and no proper extension is obtained.

"extension by localization" is owned by pahio.

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See Also: total ring of fractions, classical ring of quotients, finite ring has no proper overrings

Other names:

ring extension by localization

Also defines:

ring of fractions, ring of quotients

This object's parent.

Attachments:: total ring of fractions (Definition) by pahio

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Cross-references: finite ring, integral domain, isomorphism, ring, zero divisors, contains, units, homomorphism, well-defined, mapping, isomorphic, subring, multiplicative subset, commutative ring
There are 4 references to this entry.

This is version 12 of extension by localization, born on 2004-06-14, modified 2008-01-26.
Object id is 5916, canonical name is ExtensionByLocalization.
Accessed 3202 times total.

Classification:

13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

Pending Errata and Addenda

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