The Regular Digraph of Ideals of a Commutative Ring

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 −−→ Γreg(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 Γreg(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(Γreg(R)) |Max (R)| and χ(Γreg(R)) = 2|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 −−→ Γreg(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.