The -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 -annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let be a commutative ring and an integer greater than 2 and let be the set of all -annihilating-ideals of . The -annihilating-ideal hypergraph of , denoted by , is a hypergraph with vertex set , and for distinct elements in , the set is an edge of if and only if and the product of any elements of the 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 and characterize all commutative ring whose 3-annihilating-ideal hypergraph is planar.

Keywords

• annihilating-ideal
• hypergraph
• incidence graph
• planar
• -annihilating-ideal hypergraph