2015/10/12 by Bedanta Bose, Bose, Bedanta, Mayukh Mukherjee +1
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1510.03242
openalex publication_date 2015/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider rings of single variable real analytic or complex entire functions, denoted by \mathbbK⟨ z⟩. We study "discrete z-filters" on \mathbbK and their connections with the space of maximal ideals of \mathbbK⟨ z⟩, which we characterize as a compact T1 space θ\mathbbK of discrete z-ultrafilters on \mathbbK. We show that θ\mathbbK is a bijective continuous image of β\mathbbK ∖ Q(\mathbbK), where Q(\mathbbK) is the set of far points of β\mathbbK. θ\mathbbK turns out to be the Wallman compactification of the canonically embedded image of \mathbbK inside θ\mathbbK. Using our characterization of θ\mathbbK, we derive a Gelfand-Kolmogorov characterization of maximal ideals of \mathbbK⟨ z⟩ and show that the Krull dimension of \mathbbK⟨ z⟩ is at least c. We also establish the existence of a chain of prime z-filters on \mathbbK consisting of at least 2c many elements.