Prime Density and Classification of Mac\'ias Spaces over Principal Ideal Domains

TL;DR

This paper investigates the properties of Macías spaces over principal ideal domains, establishing conditions for prime density, unit group topology, and space homeomorphisms, extending previous results without cardinality restrictions.

Contribution

It removes the cardinality assumption in Macías topology analysis over integral domains, providing new characterizations and resolving an open problem on space homeomorphisms.

Findings

Units are not open in Macías topology for semiprimitive domains.

Equivalence between Jacobson radical triviality, prime density, and units not being open.

Complete characterization of homeomorphisms over infinite principal ideal domains.

Abstract

Recently, the Mac\'ias topology has been generalized over integral domains that are not fields, to furnish a topological proof of the infinitude of prime elements under the assumption that the set of units is finite or not open. In this article, we remove this cardinality assumption completely by using the Jacobson radical. We prove that in any semiprimitive integral domain, the group of units is not open in the Mac\'ias topology. Consequently, for a principal ideal domain, this gives an equivalence between the triviality of the Jacobson radical, the density of the set of prime elements, and the group of units not being open in the Mac\'ias topology. Furthermore, we completely characterize when Mac\'ias spaces over different infinite principal ideal domains are homeomorphic in terms of cardinalities of certain subsets of the domains. As an application we resolve an open problem concerning homeomorphism of Mac\'ias spaces over countably infinite semiprimitive principal ideal domains.