Ziegler Spectra of Serial Rings

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.