剩余类环上全矩阵环的拟零因子图性质

Properies of Quasi-Zero-Divisor Graphs of Full Matrix Rings over Z_(m)

近20年来,环论与图论相结合的零因子图一直是数学研究的热点。很多学者在环上按照一定关系定义了多种图,以此研究环的性质与图的性质之间的关系。本文研究剩余类环上全矩阵环的拟零因子图的性质,给出矩阵是剩余类环上全矩阵环的拟零因子图中顶点的充要条件,并且给出剩余类环上全矩阵环的拟零因子图中任意2个顶点的距离等于1、2、3的充要条件,最后证明2个剩余类环上全矩阵环的拟零因子图同构当且仅当全矩阵环的底环同构,且全矩阵环的阶数相同。

In the past two decades,the zero-divisor graphs that combine ring theory and graph theory have been a hot spot in mathematical research.Many scholars have defined a variety of graphs on the ring according to certain relationships,which are used to study the relationship between the properties of the rings and the properties of the graphs.This paper studies the properties of the quasi-zero divisor graphs of the full matrix rings over the residual class rings.The necessary and sufficient conditions for the matrix to be a vertex are shown in the quasi-zero divisor graphs of the full matrix rings over the residual class rings,and the necessary and sufficient conditions for any two vertices are shown in the quasi-zero divisor graphs of the full matrix rings that the distance between them is equal to 1,2,3.Finally it is proved that the quasi-zero divisor graphs of the full matrix rings over the two residual class rings are isomorphic if and only if the ground ring of the full matrix rings are isomorphic,and the order of the full matrix rings are the same.