Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (6): 116-121.doi: 10.16088/j.issn.1001-6600.2021120903

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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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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

CLC Number: 

  • O153.3
[1] BECK I. Coloring of commutative rings[J]. Journal of Algebra, 1988, 116(1): 208-226.
[2] ANDERSON D F, LIVINGSTON P S. The zero-divisor graph of a commutative ring[J]. Journal of Algebra, 1999, 217(2): 434-447.
[3] REDMOND S P. The zero-divisor graph of a non-commutative ring[J]. International Journal of Commutative Rings, 2002, 1(4): 203-211.
[4] BEHBOODI M, BEYRANVAND R. Strong zero-divisor graphs of non-commutative rings[J]. International Journal of Algebra, 2008, 2(1): 25-44.
[5] ALIBEMANI A, BAKHTYIARI M, NIKANDISH R, et al. The annihilator ideal graph of a commutative ring[J]. Journal of the Korean Mathematical Society, 2015, 52(2): 417-429.
[6] 徐承杰,易忠,郑英. 形式三角矩阵环的零因子图[J]. 数学杂志, 2013, 33(5): 891-901.
[7] 曾庆雨,易忠,唐高华,等. 矩阵环Mn(R)的中心图[J]. 西北师范大学学报(自然科学版), 2016, 52(5): 32-35.
[8] 郭述锋,黄逸飞,易忠. 群环ZnG的零因子图的性质Ⅲ[J]. 数学的实践与认识, 2019, 49(12): 278-285.
[9] 唐高华,李玉,张恒斌,等. 一类有零因子的有限环的结构[J]. 广西师范大学学报(自然科学版), 2015, 33(1): 67-73.
[10] 韦扬江, 梁艺耀,唐高华,等. 模n高斯整数环的商环的立方映射图[J]. 广西师范大学学报(自然科学版), 2016, 34(3): 53-61.
[11] 唐高华,李玉,吴严生. 群环Zn[i]G的零因子图的性质[J]. 广西师范大学学报(自然科学版), 2016, 34(4): 109-115.
[12] 韦扬江,唐高华. Zn[i]的平方映射图[J]. 数学杂志, 2016, 36(4): 676-682.
[13] 韦扬江,梁艺耀,唐高华. 高斯整数环的商环的5次幂映射图[J]. 四川师范大学学报(自然科学版), 2018, 41(2): 210-219.
[14] ZHAO S X, NAN J Z, TANG G H. Quasy-zero-divisor graphs of non-commutative rings[J]. Journal of Mathematical Research with Applications, 2017, 37(2): 135-147.
[15] BONDY J A, MURTY U S R. Graph theory[M]. New York: Springer, 2008.
[16] 华罗庚,万哲先. 华罗庚文集代数卷I[M]. 北京:科学出版社,2010.
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