The k-annihilating-ideal hypergraph of commutative ring

Abstract

The concept of the annihilating-ideal graph of a commutative ring was introduced by Behboodi et. al in 2011. In this paper, we extend this concept to the hypergraph for which we define an algebraic structure called k-annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let R be a commutative ring and k an integer greater than 2 and let A(R,k)be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by AGk(R), is a hypergraph with vertex set A(R,k), and for distinct elements I1,I2,…,Ikin A(R,k), the set {I1,I2,…,Ik}is an edge of AGk(R)if and only if ∏i=1kIi=(0)and the product of any (k−1)elements of the {I1,I2,…,Ik}is nonzero. In this paper, we provide a necessary and sufficient condition for the completeness of 3-annihilating-ideal hypergraph of a commutative ring. Further, we study the planarity of AG3(R)and characterize all commutative ring R whose 3-annihilating-ideal hypergraph AG3(R)is planar. Keywords: Annihilating-ideal, Hypergraph, Incidence graph, Planar, k-annihilating-ideal hypergraph