Prüfer domain
A commutative^{} integral domain^{} $R$ is a Prüfer domain if every finitely generated^{} nonzero ideal $I$ of $R$ is invertible.
Let ${R}_{I}$ denote the localization^{} of $R$ at $R\backslash I$. Then the following statements are equivalent:

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i) $R$ is a Prüfer domain.

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ii) For every prime ideal^{} $P$ in $R$, ${R}_{P}$ is a valuation domain.

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iii) For every maximal ideal^{} $M$ in $R$, ${R}_{M}$ is a valuation domain.
A Prüfer domain is a Dedekind domain if and only if it is Noetherian^{}.
If $R$ is a Prüfer domain with quotient field $K$, then any domain $S$ such that $R\subset S\subset K$ is Prüfer.
References
 1 Thomas W. Hungerford. Algebra^{}. SpringerVerlag, 1974. New York, NY.
Title  Prüfer domain 

Canonical name  PruferDomain 
Date of creation  20130322 13:47:34 
Last modified on  20130322 13:47:34 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  8 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 16U10 
Related topic  ValuationDomain 
Related topic  DedekindDomain 
Related topic  PruferRing 
Related topic  InvertibleIdealsInSemiLocalRings 