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 annihilatingideal 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 $mathbb{ag}_s(r)$.