Zero-divisor graph and comaximal graph of rings of continuous functions with countable range

Abstract

In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and the comaximal graph [Formula: see text] of the ring [Formula: see text] of all real-valued continuous functions having countable range, defined on any zero-dimensional space [Formula: see text], 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 [Formula: see text] of an intermediate ring [Formula: see text] of [Formula: see text] is complemented if and only if the space of all minimal prime ideals of [Formula: see text] is compact. Moreover, [Formula: see text] is complemented when and only when its subgraph [Formula: see text] is complemented. On the other hand, the comaximal graph of [Formula: see text] is complemented if and only if the comaximal graph of its over-ring [Formula: see text] is complemented and the latter graph is known to be complemented if and only if [Formula: see text] is a [Formula: see text]-space. Indeed, for a large class of spaces (i.e. for perfectly normal, strongly zero-dimensional spaces which are not P-spaces), [Formula: see text] and [Formula: see text] are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilized to establish that for a discrete space [Formula: see text], [Formula: see text] (= [Formula: see text]) and [Formula: see text] (= [Formula: see text]) are isomorphic, if [Formula: see text] is at most countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.