On the genus of the k-annihilating-ideal hypergraph of commutative rings

Abstract

Let R be a commutative ring and k an integer greater than 2 and let $${\cal A}(R, k)$$ be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by $${\cal A}{{\cal G}_k}(R)$$ , is a hypergraph with vertex set $${\cal A}(R, k)$$ , and for distinct elements I1, …, Ik in $${\cal A}(R, k)$$ , the set {I1, I2, …, Ik} is an edge of $${\cal A}{{\cal G}_k}(R)$$ if and only if $$\prod\limits_{i = 1}^k {{I_i} = (0)} $$ and the product of any (k − 1) elements of the set {I1, I2,…, Ik} is nonzero. In this paper, we characterize all Artinian commutative nonlocal rings R whose $${\cal A}{{\cal G}_3}(R)$$ has genus one.