2022/06/11 by Rakesh Bharati, Amrita Acharyya, Bharati, Rakesh +5
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2206.05471
openalex publication_date 2022/06/11 · openalex created_date 2022/06/17 · openalex updated_date 2026/07/28
In this paper, two outwardly different graphs, namely, the zero divisor graph Γ(Cc(X)) and the comaximal graph Γ2'(Cc(X)) of the ring Cc(X) of all real-valued continuous functions having countable range, defined on any Hausdorff zero dimensional space X, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness. are concerned. However, the study reveals that the zero divisor graph Γ(Ac(X)) of an intermediate ring Ac(X) of Cc(X) is complemented if and only if the space of all minimal prime ideals of Ac(X) is compact. Moreover, Γ(Cc(X)) is complemented when and only when its subgraph Γ(Ac(X)) is complemented. On the other hand, the comaximal graph of Cc(X) is complemented if and only if the comaximal graph of its over-ring C(X) is complemented and the latter graph is known to be complemented if and only if X is a P-space. Indeed, for a large class of spaces (i.e., for perfectly normal, strongly zero dimensional spaces which are not P-spaces), Γ(Cc(X)) and Γ2'(Cc(X)) are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilised to establish that for a discrete space X, Γ(Cc(X)) (= Γ(C(X))) and Γ2'(Cc(X)) (= Γ2'(C(X))) are isomorphic, if X is atmost countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.