素数密度与主理想域上Macías空间的分类
Prime Density, Kolmogorov Quotients and Homeomorphisms between Macías Spaces of Integral Domains
AI总结：
本文研究了主理想域上Macías空间的素数密度与单位群性质，通过Jacobson根理论去除基数假设，证明了单位群在Macías拓扑下不开放，并解决了Macías空间的homeomorphism问题。
摘要：
近年来，Macías拓扑已被推广到非域的整环上，以提供素元无穷性的拓扑证明，前提是单位集有限或非开放。本文通过使用Jacobson根完全去除了这个基数假设。我们证明在任何半素整环中，单位群在Macías拓扑下不开放。因此，在主理想域中，这等价于Jacobson根的平凡性、素元集的密度性和单位群在Macías拓扑下不开放。此外，我们完全刻画了不同无限主理想域上Macías空间的homeomorphism条件，以基数为特征。作为应用，我们解决了关于计数无限半素主理想域上Macías空间homeomorphism的开放问题。
英文摘要：
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 $σ_k^0=\{x\in R^0:\langle x\rangle+\langle k\rangle=R\}$ 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.
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