On the arithmetic of stable domains

TL;DR

This paper investigates the arithmetic properties of stable integral domains, especially focusing on the semigroups of ideals in stable orders within Dedekind domains, building on classical algebraic theory.

Contribution

It provides new insights into the arithmetic structure of stable domains and their ideal semigroups, extending classical results in the context of Dedekind domains.

Findings

Characterization of ideal semigroups in stable orders

Connections between stability and divisoriality

New properties of stable integral domains

Abstract

A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the $2$-generator property. In the present paper we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.