Descending chains of semistar operations

TL;DR

This paper introduces ideal valuations as a new way to understand chains of semistar operations, linking them to classical operations and solving a conjecture in valuation domains.

Contribution

It establishes a one-to-one correspondence between ideal valuations and descending chains of semistar operations, providing new insights and solutions in the theory of integral domains.

Findings

Ideal valuations correspond to chains of semistar operations.

The approach recovers known operations like the w-operation.

A conjecture by Chapman and Glaz is solved for valuation domains.

Abstract

A class of integer-valued functions defined on the set of ideals of an integral domain $R$ is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the $e$-operation. We use this class of functions to recover familiar semistar operations such as the $w$-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.