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 $\Gamma(C_c(X))$ and the comaximal graph $\Gamma_2^{'}(C_c(X))$ of the ring $C_c(X)$ 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 $\Gamma(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the space of all minimal prime ideals of $A_c(X)$ is compact. Moreover, $\Gamma(C_c(X))$ is complemented when and only when its subgraph $\Gamma(A_c(X))$ is complemented. On the other hand, the comaximal graph of $C_c(X)$ 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), $\Gamma(C_c(X))$ and $\Gamma_2^{'}(C_c(X))$ are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilised to establish that for a discrete space $X$, $\Gamma(C_c(X))$ (= $\Gamma(C(X))$) and $\Gamma_2^{'}(C_c(X))$ (= $\Gamma_2^{'}(C(X))$) are isomorphic, if $X$ is atmost countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.