# 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 ξR, 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 Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basic
properties of the graph ξR. In particular, we show
that ξR is a connected graph with diameter at most three, and
has girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

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Main Subjects