Radical semistar operations

TL;DR

This paper introduces and analyzes radical stable semistar operations on integral domains, revealing their lattice structure, spectral conditions, and classifications in specific domain types like Pr"ufer domains.

Contribution

It characterizes the lattice of radical stable operations, relates them to spectral operations, and classifies stable semistar operations in Pr"ufer domains with scattered minimal primes.

Findings

The set of radical stable operations forms a complete lattice.

Conditions are provided for when all radical operations are spectral.

Complete classification of stable semistar operations in certain Pr"ufer domains.

Abstract

We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that $D$ is rad-colon coherent). When $D$ is a Pr\"ufer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.