Zero-Divisor Graph of Real-Valued Continuous Functions on a Frame

Abstract

The main object of this paper is to study the zero-divisor graph $\Gamma(\mathcal{R}L)$ of the ring $\mathcal{R}L$. We have communicated the ring-theoretic properties of $\mathcal{R}L$, the graph-theoretic properties of $\Gamma(\mathcal{R}L)$ and the pointfree-theoretic properties of $L$. Paths in $\Gamma(\mathcal{R}L)$ are investigated, and it is shown that the diameter of $\Gamma(\mathcal{R}L)$ and the girth of $\Gamma(\mathcal{R}L)$ coincide whenever $L$ has at least $5$ elements. Cycles in $\Gamma(\mathcal{R}L)$ are surveyed and a ring-theoretic and a pointfree-theoretic characterization is provided for the graph $\Gamma(\mathcal{R}L)$ to be triangulated or be hypertriangulated. We show that $\Gamma(\mathcal{R}L)$ is complemented if and only if the space of minimal prime ideals of $\mathcal{R}L$ is compact. The relation between the clique number of $\Gamma(\mathcal{R}L)$, the cellularity of $L$ and the dominating number of $\Gamma(\mathcal{R}L)$ is given. Finally we prove that $\Gamma(\mathcal{R}L)$ is not triangulated and the set of centers of $\Gamma(\mathcal{R}L)$ elements is a dominating set if and only if the socle of $\mathcal{R}L$ is an essential ideal.