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

Let R be a commutative ring with identity and 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 AGP(R). It is a (undirected) graph with vertices AP(R)=A(R)∩P(R)∖{(0)}, where 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=(0). Then, we study some basic properties of AGP(R). For instance, we characterize rings for which AGP(R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of AGP(R). Finally, we compare the principal ideal subgraph AGP(R) and spectrum subgraph AGs(R).