Boundness in almost Dedekind domains

TL;DR

This paper investigates various forms of boundness in almost Dedekind domains, revealing structural properties such as the density of noncritical maximal ideals and the freeness of the invertible ideal group.

Contribution

It generalizes notions of critical ideals and radical factorization to almost Dedekind domains, establishing new properties like SP-scatteredness and the existence of bounded elements.

Findings

Noncritical maximal ideals are dense in the maximal space.

Almost Dedekind domains are SP-scattered.

The group of invertible ideals is always free.

Abstract

We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal ideals and, indeed, the set of noncritical maximal ideals is dense in the maximal space, with respect to the constructible topology; as a consequence, we show that every almost Dedekind domain is SP-scattered, and in particular that the group $\mathrm{Inv}(D)$ of invertible ideals of an almost Dedekind domain $D$ is always free. If $D$ is an almost Dedekind domain with nonzero Jacobson radical, we also show that there is at least one element whose ideal function is bounded.