Ziegler Spectra of Serial Rings

TL;DR

This paper proves the sobriety of Ziegler spectra for serial rings and introduces a framework for their computation, demonstrated through rank one uniserial domains.

Contribution

It establishes the sobriety of Ziegler spectra for serial rings and develops a method for their analysis and computation.

Findings

Ziegler spectra of serial rings are sober.

A general framework for computing Ziegler spectra of uniserial rings is provided.

Explicit computation of spectra for rank one uniserial domains is achieved.

Abstract

In this paper we prove that the Ziegler spectra of all serial rings are sober. We then use this proof to give a general framework for computing and understanding Ziegler spectra of uniserial rings up to topological indistinguishability. Finally, we illustrate this technique by computing the Ziegler spectra of all rank one uniserial domains up to topological indistinguishability.