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:

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

单词	PruferDomain
释义	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.
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