The Classification of the Annihilating-Ideal Graphs of Commutative Rings

• 作者: F. Shaveisi G. Aalipour M. Behboodi M.J. Nikmehr R. Nikandish S. Akbari
• 来源期刊: 代数集刊:英文版
• 学科: 数学
• 关键词: 交换环 理想 分类 顶点集 零化子 毁灭 环和 Ag
• 年，卷（期）: 2014,（2）
• 页码范围: 249-256
• 页数: 8页
• 分类号: O153.3
• 语种: 中文

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.