Towards a classification of stable semistar operations on a Pr\"ufer domain

TL;DR

This paper investigates stable semistar operations on Pr"ufer domains, demonstrating that under certain conditions, these operations can be characterized via valuation overrings, advancing the understanding of ideal closures in such domains.

Contribution

It provides a classification of stable semistar operations on Pr"ufer domains with finitely many minimal primes per ideal, linking them to valuation overrings.

Findings

Stable semistar operations can be described via valuation overrings.

Pr"ufer domains with finitely many minimal primes per ideal admit a specific classification.

The work extends the understanding of ideal closure operations in Pr"ufer domains.

Abstract

We study stable semistar operations defined over a Pr\"ufer domain, showing that, if every ideal of a Pr\"ufer domain $R$ has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of $R$.