We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the P-adic topology on R∖P. Under certain hypothesis, we show that we can associate to a prime ideal P of R a partially ordered set, constructed from some subgroups of the group of units of R/Pn, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of Z are the identity and the multiplication by −1. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of Q and different from Z is not homeomorphic to the Golomb space of Z.

The Golomb topology on a Dedekind domain and the group of units of its quotients

Spirito D.
2020

Abstract

We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the P-adic topology on R∖P. Under certain hypothesis, we show that we can associate to a prime ideal P of R a partially ordered set, constructed from some subgroups of the group of units of R/Pn, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of Z are the identity and the multiplication by −1. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of Q and different from Z is not homeomorphic to the Golomb space of Z.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11390/1215959
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