Prime Density and Classification of Mac\'ias Spaces over 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.