The annihilator ideal graph of a commutative ring

Abstract

Let $R$ be a commutative ring with nonzero identity and $I$ be a proper ideal of $R$. The annihilator graph of $R$ with respect to $I$, which is denoted by $AG_{I}(R)$, is the undirected graph with vertex-set $V(AG_{I}(R)) = \lbrace x\in R \setminus I : xy \in I\ $ for some$ \ y \notin I \rbrace$ and two distinct vertices $x$ and $y$ are adjacent if and only if $A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y)$, where $A_{I}(x) = \lbrace r\in R : rx\in I\rbrace$. In this paper, we study some basic properties of $AG_I(R)$, and we characterise when $ AG_{I}(R) $ is planar, outerplanar or a ring graph. Also, we study the graph $AG_{I}(\mathbb{Z}_{n}) $, where $Z_n$ is the ring of integers modulo $n$.