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Totally ordered sets and the prime spectra of rings. (English) Zbl 1387.16002

Summary: Let \(T\) be a totally ordered set and let \(D(T)\) denotes 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 \(\mathrm{Spec}(O_v)\). 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 \(U\subseteq\mathrm{Spec}(A)\) 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.

MSC:

16D25 Ideals in associative algebras
13A18 Valuations and their generalizations for commutative rings
06A05 Total orders
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