# The annihilator-inclusion Ideal graph of a commutative ring

Document Type : Original paper

Authors

2 Jabir Ibn Hayyan research center

Abstract

Let $R$ be a commutative ring with non-zero identity.   The  annihilator-inclusion ideal graph of $R$, denoted by $\xi_R$, is a graph whose vertex set is the of all non-zero proper ideals of $R$  and two distinct vertices $I$ and $J$ are adjacent if and only if either ${\rm Ann}(I)\subseteq J$ or ${\rm Ann}(J)\subseteq I$.  The purpose of this paper is to provide some basic properties of the graph $\xi_R$. In particular, shows that $\xi_R$ is a connected graph with diameter at most three, and has girth 3 or $\infty$.   Furthermore,  is  determined all isomorphic classes of non-local Artinian  rings whose annihilator-inclusion ideal graphs have genus zero or one.

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