Totally ordered sets and the prime spectra of rings.

LetTbe a totally ordered set and letD(T) denotes the set of all cuts ofT. We prove the existence of a discrete valuation domainOvsuch thatTis order isomorphic to two special subsets of Spec(Ov). We prove that ifAis a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered setU⊆Spec(A) such that the prime spectrum ofAis order isomorphic toD(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. [ABSTRACT FROM AUTHOR]