Let R be a commutative ring and 𝔸 (R) be the set of ideals with nonzero annihilators. The
annihilating-ideal graph of R is defined as the graph 𝔸𝔾 (R) with the vertex set 𝔸 (R)*=
𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ=(0). We investigate
commutative rings R whose annihilating-ideal graphs have positive genus γ (𝔸𝔾 (R)). It is
shown that if R is an Artinian ring such that γ (𝔸𝔾 (R))<∞, then either R has only finitely
many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v. dim R/𝔪𝔪/𝔪2= 2 …

Let R be a commutative ring with 1= 0 and let I (R) be the set of all proper ideals of R. An
ideal I∈ I (R) is called an annihilator ideal of R if, IJ= 0 for some nonzero ideal J∈ I (R). Let
A (R) denote the set of all annihilators ideals of R. In this paper, we define the Annihilating-
Ideal graph of R (denoted by AG (R)), as an undirected graph with vertices A (R)∗= A
(R)\{(0)}, where distinct vertices I and J are adjacent if and only if IJ= 0. We investigate
commutative rings whose annihilating-ideal graphs have positive genus. It is shown that if R …