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.

Bibliography

1

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