Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

Abstract

In this article we introduce the zero-divisor graphs $\Gamma_\mathscr{P}(X)$ and $\Gamma^\mathscr{P}_\infty(X)$ of the two rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$; here $\mathscr{P}$ is an ideal of closed sets in $X$ and $C_\mathscr{P}(X)$ is the aggregate of those functions in $C(X)$, whose support lie on $\mathscr{P}$. $C^\mathscr{P}_\infty(X)$ is the $\mathscr{P}$ analogue of the ring $C_\infty (X)$. We find out conditions on the topology on $X$, under-which $\Gamma_\mathscr{P}(X)$ (respectively, $\Gamma^\mathscr{P}_\infty(X)$) becomes triangulated/ hypertriangulated. We realize that $\Gamma_\mathscr{P}(X)$ (respectively, $\Gamma^\mathscr{P}_\infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_\mathscr{P}(X)$ (respectively $\Gamma^\mathscr{P}_\infty(X)$) is compact. This places a special case of this result with the choice $\mathscr{P}\equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in \cite{Azarpanah} on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals $\mathscr{P}$ and $\mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_\mathscr{P}(X)$ and $C_\mathscr{Q}(Y)$ are isomorphic if and only if $\Gamma_\mathscr{P}(X)$ and $\Gamma_\mathscr{Q}(Y)$ are isomorphic.