# 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.

## References

• 1 Thomas W. Hungerford. Algebra. Springer-Verlag, 1974. New York, NY.
Title Prüfer domain PruferDomain 2013-03-22 13:47:34 2013-03-22 13:47:34 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 16U10 ValuationDomain DedekindDomain PruferRing InvertibleIdealsInSemiLocalRings