群环Z_nG的零因子图的性质,Ⅲ

Properties of Zero-divisor Graphs of Group Rings Z_nG,Ⅲ

环的零因子图是20世纪90年代才兴起的一个数学研究方向.环上的零因子图的研究,刻画了环的零因子的结构,这对理解环结构本身具有重要意义.群环是群论和环论的交汇点之一.对它的研究在环论,群论及伽罗华理论等学科领域都有重要的意义.主要讨论了群环ZnG的零因子图的性质,对群环ZnG的零因子图的围长,平面性,直径给出了较为具体的刻画,其中G为非循环的有限交换群.

The zero-divisor graph of a ring is a mathematical research direction that rose in 1990’s.The study of zero-divisor graphs of rings is devoted to characterizing the structure of their zero-divisors.It is important to understand the structure of rings.The group ring is one of the intersections of group theory and ring theory.It is of great significance to study it in the ring theory,group theory and Galois theory.Let G be a acyclic finite abelian group and ZnG group rings of G over Zn.Properties of zero-divisor graphs of ZnG are mainly discussed in this paper.And the characterizations on the girth,the diameter and the planarity of zero-divisor graphs of ZnG are given respectively.