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 $mathbb{A}_P(R)=mathbb{A}(R)cap mathbb{P}(R)setminus {(0)}$, 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 $IJ=(0)$. 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 $mathbb{AG}_s(R)$.