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Abstract
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Let R be a commutative ring with non-zero identity. The annihilator-
inclusion ideal graph of R, denoted by 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 Ann(I) J or Ann(J) I. The purpose of this paper is to provide some
basic properties of the graph R. In particular, shows that R is a connected graph
with diameter at most three, and has girth 3 or 1. 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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