The regular digraph of ideals of a commutative ring

Abstract

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by $\overrightarrow{\Gamma_{\mathrm{reg}}}(R)$ , is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γ_reg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max (R)|−1≦ω(Γ_reg(R))≦|Max (R)| and $\chi(\Gamma_{\mathrm{ reg}}(R)) = 2|\mathrm{Max}\, (R)| -k-1$ , where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that $\overrightarrow{\Gamma_{\mathrm{ reg}}}(R)$ is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.