广西师范大学学报（自然科学版） ›› 2022, Vol. 40 ›› Issue (6): 116-121.doi: 10.16088/j.issn.1001-6600.2021120903

• 研究论文 • 上一篇    下一篇

剩余类环上全矩阵环的拟零因子图性质

赵寿祥1,2, 唐高华3*, 南基洙1   

  1. 1.大连理工大学数学科学学院,辽宁大连116024;
    2.桂林师范高等专科学校数学与计算机技术系,广西桂林541199;
    3.北部湾大学理学院,广西钦州535011
  • 收稿日期:2021-12-09 修回日期:2022-02-24 出版日期:2022-11-25 发布日期:2023-01-17
  • 通讯作者: 唐高华(1965—),男,广西桂林人,北部湾大学教授,博导。E-mail:tanggaohua@163.com
  • 基金资助:
    国家自然科学基金(11961050,12171194);广西高校中青年教师科研基础能力提升项目(2019KY1161)

Properies of Quasi-Zero-Divisor Graphs of Full Matrix Rings over Zm

ZHAO Shouxiang1,2, TANG Gaohua3*, NAN Jizhu1   

  1. 1. School of Mathematical Sciences, Dalian University of Technology, Dalian Liaoning 116024, China;
    2. Department of Mathematics and Computer Technology, Guilin Normal College, Guilin Guangxi 541199, China;
    3. School of Sciences, Beibu Gulf University, Qinzhou Guangxi 535011, China
  • Received:2021-12-09 Revised:2022-02-24 Online:2022-11-25 Published:2023-01-17

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摘要： 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。

关键词: 零因子图, 拟零因子图, 全矩阵环, 剩余类环, 图的直径

Abstract: 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.

Key words: zero divisor graph, quasi-zero-divisor graph, full matrix ring, residue class ring, diameter of graphs

中图分类号: 

  • O153.3
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[3] 郭述锋, 谢光明, 易忠. 群环ZnG的零因子图的性质[J]. 广西师范大学学报（自然科学版）, 2015, 33(2): 68-75.
[4] 黄逸飞, 易忠, 覃庆玲. 群环ZnD4的零因子图[J]. 广西师范大学学报（自然科学版）, 2011, 29(2): 15-20.
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