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:

• i) $R$ is a Prüfer domain.

• ii) For every prime ideal $P$ in $R$, $R_P$ is a valuation domain.

• 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.

Title	Prüfer domain
Canonical name	PruferDomain
Date of creation	2013-03-22 13:47:34
Last modified on	2013-03-22 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