The Classification of the Annihilating-Ideal Graphs of Commutative Rings  

作　　者：G. Aalipour S. Akbari M. Behboodi R. Nikandish M.J. Nikmehr F. Shaveisi 

机构地区：[1]Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran [2]School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Iran [3]Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran [4]Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran

出　　处：《Algebra Colloquium》2014年第2期249-256,共8页代数集刊（英文版）

摘　　要：Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

关 键 词：annihilating-ideal graph clique number chromatic number Artinian ring Noetherian ring 

分 类 号：O153.3[理学—数学] O157.5[理学—基础数学]