Prüfer domain

Remarks

Equivalent form: for every prime ideal $P$ of $R$, the localization $R_P$ is a valuation ring, that is, for every nonzero $x \in K$, $x \in R_P$ or $x^{-1} \in R_P$. Example: every principal ideal domain is Prüfer; a nonzero ideal $aR$ has inverse $a^{-1}R$. Non-example: in $\Z[x]$, the ideal $I = (2, x)$ is not invertible; the only $y \in \Q(x)$ with $yI \subseteq \Z[x]$ are the elements of $\Z[x]$, so any $J$ with $IJ \subseteq \Z[x]$ satisfies $IJ \subseteq I \ne \Z[x]$. A Noetherian Prüfer domain is a Dedekind domain; a Prüfer domain that is a GCD domain is a Bézout domain. Prüfer domains are integrally closed.