Prime Density, Kolmogorov Quotients and Homeomorphisms between Macías Spaces of Integral Domains

Abstract

For an integral domain $R$ that is not a field, the Macías space $M(R)$ is the set $R^0=R\setminus \{0\}$ endowed with the topology generated by the subsets ${\sigma }_k^0=\{x\in R^0:⟨x⟩+⟨k⟩=R\}$ for $k\in R^0$. We show that the unit group is open in $M(R)$ precisely when $R$ is not semiprimitive, and characterize semiprimitive Dedekind domains and unique factorization domains (UFDs) by means of the density of their irreducible and prime elements, respectively. We describe the Kolmogorov quotient of $M(R)$ in terms of the maximal ideals of $R$, and reduce the homeomorphism problem to these quotients and the cardinalities of the fibers of the quotient maps. Consequently, two local domains have homeomorphic Macías spaces if and only if they have the same cardinality. We characterize the pairs of Dedekind domains with homeomorphic Macías spaces in terms of the cardinalities of their unit groups and the relations among the ideal classes of their maximal ideals; in particular, two Dedekind domains with torsion class groups, such as principal ideal domains (PIDs), have homeomorphic Macías spaces if and only if their unit groups have the same cardinality and so do their sets of maximal ideals. This characterization addresses, for countably infinite semiprimitive PIDs, the homeomorphism problem raised by Macías in 2024. Under an additional hypothesis, UFDs admit an analogous characterization in terms of comaximality with prime elements. Finally, for a class of domains of characteristic zero, we characterize additive homeomorphisms between Macías spaces in terms of isomorphisms of the underlying rings, partially answering a problem raised by Macías and Ortiz in 2024.

arXiv comment24 pages. Comments are welcome