PlanetMath: total ring of fractions

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total ring of fractions

(Definition)

For a commutative ring $R$ having regular elements, we may form $T = S^{-1}R$ the total ring of fractions (quotients) of $R$ as the localization of $R$ at $S$ where $S$ is the set of all non-zero-divisors of $R$ Then, $T$ can be regarded as an extension ring of $R$ (similarly as the field of fractions of an integral domain is an extension ring). $T$ has the non-zero unity 1.

"total ring of fractions" is owned by pahio.

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Other names:

total ring of quotients

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Attachments:: overring (Definition) by pahio
generators of inverse ideal (Theorem) by pahio
quotient of ideals (Definition) by pahio
fractional ideal of commutative ring (Definition) by pahio

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Cross-references: non-zero unity, integral domain, field of fractions, ring, extension, localization, quotients, regular elements, commutative ring
There are 12 references to this entry.

This is version 10 of total ring of fractions, born on 2004-05-21, modified 2005-04-26.
Object id is 5866, canonical name is TotalRingOfFractions.
Accessed 4066 times total.

Classification:

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

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