ON THE REGULAR DIGRAPH OF IDEALS OF COMMUTATIVE RINGS

Let R be a commutative ring. The regular digraph of ideals of R, denoted by Gamma(R), is a digraph whose vertex set is the set of all nontrivial ideals of R and, for every two distinct vertices I and J, there is an arc from I to J whenever I contains a nonzero divisor on J. In this paper, we study the connectedness of Gamma(R). We also completely characterise the diameter of this graph and determine the number of edges in Gamma(R), whenever R is a finite direct product of fields. Among other things, we prove that R has a finite number of ideals if and only if N-Gamma(R)(I) is finite, for all vertices I in Gamma(R), where N-Gamma(R)(I) is the set of all adjacent vertices to I in Gamma(R).