The principal ideal subgraph of the annihilating-ideal graph of commutative rings

Let R be a commutative ring with identity and \mathbb{A}(R) be the set   of ideals of R with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of R, denoted by \mathbb{AG}_P(R). It is a (undirected) graph with vertices \mathbb{A}_P(R)=\mathbb{A}(R)\cap \mathbb{P}(R)\setminus \{(۰)\}, where   \mathbb{P}(R) is the set of  proper principal ideals of R and two distinct vertices I and J are adjacent if and only if IJ=(۰). Then, we study some basic properties of \mathbb{AG}_P(R). For instance, we characterize rings for which \mathbb{AG}_P(R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of \mathbb{AG}_P(R). Finally, we compare  the principal ideal subgraph \mathbb{AG}_P(R) and spectrum subgraph \mathbb{AG}_s(R).

Keywords:

commutative rings , annihilating-ideal , principal ideal , graph