Almost Dedekind domains without radical factorization

TL;DR

This paper investigates the structure of almost Dedekind domains focusing on their failure to have radical factorizations, introducing SP-scattered domains and analyzing their ideal groups and length functions.

Contribution

It introduces the concept of SP-scattered domains, generalizes critical ideals, and characterizes the ideal group structure and length functions for these domains.

Findings

The group of invertible ideals is free for SP-scattered domains.

Length of R/I equals length of R/rad(I) for these domains.

Provides a framework to measure how far an almost Dedekind domain is from being an SP-domain.

Abstract

We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space $\mathcal{M}=\mathrm{Max}(R)$ of an almost Dedekind domain $R$, interpreting its (fractional) ideals as maps from $\mathcal{M}$ to $\mathbb{Z}$, and looking at the continuity of these maps when $\mathcal{M}$ is endowed with the inverse topology and $\mathbb{Z}$ with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of $\mathcal{M}$ (of which the set of critical ideals is the first step) and use it to define the class of \emph{SP-scattered domains}, which includes the almost Dedekind domains such that $\mathcal{M}$ is scattered and, in particular, the almost Dedekind domains such that $\mathcal{M}$ is countable. We show that for this class of rings the group $\mathrm{Inv}(R)$ is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function $\ell$ on $R$ and every ideal $I$ of $R$, the length of $R/I$ is equal to the length of $R/\mathrm{rad}(I)$.