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

TL;DR

This paper introduces zero-divisor graphs for certain rings of functions on topological spaces, exploring their properties and conditions for graph isomorphisms and special topological features.

Contribution

It defines new zero-divisor graphs for rings of functions with support on specific ideals, analyzing their properties and establishing conditions for graph and ring isomorphisms.

Findings

Zero-divisor graphs are triangulated or hypertriangulated under certain topological conditions.

Complemented graphs correspond to compactness of minimal prime ideal spaces.

Ring isomorphism is characterized by graph isomorphism for specific ideals.

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.