When is an integrally closed local domain a valuation ring? Every valuation ring is an integrally closed local domain, and the integral closure of a local ring is the intersection of all valuation rings containing it. It would be useful for me to know when integrally closed local domains are valuation rings. To be more specific, > is there a property $P$ of unitary commutative rings that is strictly weaker than being a valuation ring, such that an integrally closed local domain is a valuation ring iff it satisfies the property $P$.

A commutative ring $R$ is called coherent, if every finitely generated ideal $I$ is finitely presented, that is as an $R$-module $I$ is isomorphic to $R^n/J$ for some finitely generated $R$-submodule $J$ of $R^n$.

For two ideals $I,J$ of $R$ one defines the ideal $(I:J):=\\{r\in R : rJ\subseteq I\\}$.

Now the following is true: the local integrally closed domain $R$ is a valuation domain if and only if $R$ is coherent and there exist $r,s\in R$, $s\
ot\in rR$ such that the maximal ideal $M$ of $R$ is minimal among the prime ideals containing $(rR:sR)$.

This follows from results obtained by J. Mott and M. Zafrullah some decades ago.

References:

S. Glaz, Commutative coherent rings, Lecture notes in mathematics 1371, 1989. (general theory of coherence)

J. Mott, M. Zafrullah, On Prüfer -v-multiplication domains, Manuscripta Mathematica 35 (1981). (Theorem 3.2 is relevant)

M. Zafrullah, On finite conductor domains, Manuscripta Mathematica 24 (1978). (Theorem 2 is relevant)