Totally ordered sets and the prime spectra of rings

TL;DR

This paper explores the structure of totally ordered sets and their relation to prime spectra of rings, establishing isomorphisms with spectra of valuation domains and characterizing Dedekind totally ordered sets.

Contribution

It demonstrates that totally ordered sets can be represented as prime spectra of valuation domains and characterizes Dedekind totally ordered sets.

Findings

Existence of valuation domains with spectra order isomorphic to given totally ordered sets.

Prime spectra of rings satisfying certain conditions are order isomorphic to sets of cuts.

Characterization of Dedekind totally ordered sets.

Abstract

Let $T$ be a totally ordered set and let $D(T)$ denote the set of all cuts of $T$. We prove the existence of a discrete valuation domain $O_{v}$ such that $T$ is order isomorphic to two special subsets of Spec$(O_{v})$. We prove that if $A$ is a ring (not necessarily commutative) whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set $U \subseteq \text{Spec}(A)$ such that the prime spectrum of $A$ is order isomorphic to $D(U)$. We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view