A realization theorem for almost Dedekind domains

TL;DR

This paper demonstrates that for any ordinal number, there exists an almost Dedekind domain with that specific SP-rank, expanding understanding of the structure of these domains.

Contribution

It establishes that every ordinal can be realized as the SP-rank of an almost Dedekind domain, providing a comprehensive realization theorem.

Findings

Every ordinal number can be realized as the SP-rank of an almost Dedekind domain.

The SP-rank measures the deviation of an almost Dedekind domain from being an SP-domain.

The paper characterizes the possible SP-ranks within the class of almost Dedekind domains.

Abstract

An integral domain $D$ is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of $D$ provides a natural measure of the deviation of $D$ from being an SP-domain. In the present paper we show that every ordinal number $\alpha$ can be realized as the SP-rank of an almost Dedekind domain.