期刊名称：AKCE International Journal of Graphs and Combinatorics
印刷版ISSN：0972-8600
出版年度：2019
卷号：16
期号：3
页码：241-252
DOI：10.1016/j.akcej.2019.02.008
语种：English
出版社：Elsevier
摘要：AbstractThe 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 calledk-annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. LetRbe a commutative ring andkan integer greater than 2 and letA(R,k)be the set of allk-annihilating-ideals ofR. Thek-annihilating-ideal hypergraphofR, denoted byAGk(R), is a hypergraph with vertex setA(R,k), and for distinct elementsI1,I2,…,IkinA(R,k), the set{I1,I2,…,Ik}is an edge ofAGk(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 ofAG3(R)and characterize all commutative ringRwhose 3-annihilating-ideal hypergraphAG3(R)is planar.
