Cayley Graphs of Ideals in a Commutative Ring

Let R be a commutative ring. We associate a digraph to the ideals of R 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, denoted by I -> J, whenever there exists a nontrivial ideal L such that J = IL. We call this graph the ideal digraph of R and denote it by (1 Gamma) over right arrow (R). Also, for a semigroup H and a subset S of H, the Cayley graph Cay(H,S) of H relative to S is defined as the digraph with vertex set H and edge set E(H,S) consisting of those ordered pairs (x,y) such that y = sx for some s is an element of S. In fact the ideal digraph (I Gamma) over right arrow (R) is isomorphic to the Cayley graph Cay(J*,J*), where J is the set of all ideals of R and J* consists of nontrivial ideals. The undirected ideal (simple) graph of R, denoted by I Gamma(R), has an edge joining I and J whenever either J = IL or I = JL, for some nontrivial ideal L of R. In this paper, we study some basic properties of graphs (I Gamma) over right arrow (R) and I Gamma(R) such as connectivity, diameter, graph height, Wiener index and clique number. Moreover, we study the Hasse ideal digraph (I Gamma) over right arrow (R), which is a spanning subgraph of (I Gamma) over right arrow (R) such that for each two distinct vertices I and J, there is an arc from I to J in (I Gamma) over right arrow (R) whenever I -> J in (I Gamma) over right arrow (R), and there is no vertex L such that I -> L and L -> J in (I Gamma) over right arrow (R).