PlanetMath: Prüfer domain

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Prüfer domain

(Definition)

A commutative integral domain is a Prüfer domain if every finitely generated nonzero ideal of is invertible.

Let denote the localization of at $R\backslash I$ . Then the following statements are equivalent:

i) is a Prüfer domain.
ii) For every prime ideal in , is a valuation domain.
iii) For every maximal ideal in , is a valuation domain.

A Prüfer domain is a Dedekind domain if and only if it is Noetherian.

If is a Prüfer domain with quotient field , then any domain such that $R\subset S\subset K$ is Prüfer.

Bibliography

1: Thomas W. Hungerford. Algebra. Springer-Verlag, 1974. New York, NY.

"Prüfer domain" is owned by mathcam.

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See Also: valuation domain, Dedekind domain, Prüfer ring

Keywords:

valuation domain, Prüfer, Noetherian

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Cross-references: quotient field, Noetherian, Dedekind domain, maximal ideal, valuation domain, prime ideal, equivalent, localization, invertible, ideal, finitely generated, integral domain, commutative
There are 6 references to this entry.

This is version 5 of Prüfer domain, born on 2003-07-25, modified 2004-12-07.
Object id is 4507, canonical name is PruferDomain.
Accessed 2543 times total.

Classification:

16U10 (Associative rings and algebras :: Conditions on elements :: Integral domains)

Pending Errata and Addenda

None.[ View all 4 ]

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