Z_n[ω]与形式三角矩阵环的零因子图
摘要
环论与图论是数学中的两个非常重要的分支,它们不仅内涵丰富,而且在许多其它数学分支(如组合数学、几何学、自动机理论以及编码理论等)中也有重要应用.环的零因子图,主要是使用图性质研究代数系统,它提供了一种研究数学问题的新方法.环(或群)的零因子图是最近二十年来才产生的一个新型研究领域,引发出了很多有趣的结果和问题.近十年来,它已成为国际上的一个热门研究领域.
Z [i]和Z [ω]是抽象代数中环论的两个重要的环,常被作为例子散见于各类抽象代数教材及论文中,本文讨论了代数整数环Z [ω]的模n剩余类环Z n[ω]的零因子集合、单位乘群、素谱以及Zn [ω]J的局部环直和分解,并在这些结果的基础上,研究了Z n[ω]的零因子图的直径、平面性及围长;本文还讨论了形式三角矩阵环的无向零因子图的直径,并简要地研究了任意含幺交换环上二阶上三角矩阵环的零因子图.具体地说,全文共分为四章:第一章介绍了环的零因子图的发展历史,本文的研究背景,理论来源和研究意义.并给出了本文所用到的一些基本概念与符号.
第二章主要讨论了代数整数环Z [ω]的模n剩余类环Z n[ω]的零因子集合、单位乘群、素谱以及Zn [ω]J的局部环直和分解(定理2.2.3,定理2.2.4),这些结果对我们后面的研究是非常有用的.
第三章主要是探讨有限含幺交换环的零因子图的若干图论性质,从而直观的反映环的零因子的内部结构.在第二章的基础上,讨论了有限交换环Z n[ω]的零因子图的直径(定理3.1.2)、平面性(定理3.2.4)与围长(定理3.3.2),本章的主要结果即将在湖南工业大学学报2009年第2期发表.
第四章主要讨论一类非交换环即形式三角矩阵环的无向零因子图.形式三角矩阵环是一类非常重要的非交换环,在环论的许多教材及文献中,它经常被作为反例出现.本章第一节中,在M为R无扰模的条件下,我们给出了形式三角矩阵环的零因子集合;本章第二节,我们对形式三角矩阵环的无向零因子图的直径做了一些探讨(定理4.2.4,定理4.2.6,定理4.2.9),定理4.2.4证明了任意形式三角矩阵环的零因子图直径只能为2或3.紧接着,定理4.2.6在M为R无扰模的条件下,给出了任意形式三角矩阵环的零因子图直径为2的一个必要条件,而定理4.2.9则在某一前提下,为我们提供了形式三角矩阵环的无向零因子图直径为2的一个充要条件,这三个定理对于形式三角矩阵环的零因子图的研究是非常有意义的.在这一章的最后一节中,我们利用本章前两节的结论讨论了任意含幺交换环上二阶上三角矩阵环的无向零因子图的直径(命题4.3.2-命题4.3.6),得到了本文的最后一个定理,即任意含幺交换环上二阶上三角矩阵环的无向零因子图的直径为3的一个判定定理(定理4.3.7).
文章的最后部分是本文的结束语,一方面,我们总结了本文的主要工作并介绍了本文的若干后续工作;另一方面,我们阐述了作者关于环的零因子图的几个十分感兴趣的问题.
Ring Theory and Graph Theory are two very important mathematical branches, which are not only of great theoretical interest in themselves but also found important applications in many other branches of math (such as combinatorial mathematics, geometry, automata theory and coding theory, etc.). The zero divisor graphs of rings, using properties of graphs to study algebraic structures, has become an exciting research topic in the last twenty years, leading to many fascinating results and questions. In the past ten years, it has become a hot research field.
Z [i] and Z [ω] are two important rings in Abstract Algebra, often as scattered examples in materials and thesis of various types of abstract algebra. In this paper, at first, we discuss the set of zero divisors, units group, prime spectrum of Z n[ω], and the local ring Direct Sum decomposition of Zn [ω]J. On the basis of these results, we give the diameter, the girth and planarity of the zero divisor graphs of Z n[ω]. In this paper, we also study the zero divisor graphs of formal triangular matrix rings, and briefly study the zero divisor graphs of 2×2 upper triangular matrix rings. Specifically, the text is divided into four chapters:
In the first chapter, we summarize the history of the zero-divisor graph, and the background of this paper. At the same time, we give some notations and basic definitions of ring theory and graph theory.
In the second chapter, we discuss the set of zero-divisors , units group, prime spectrum of Z n[ω] and the local ring Direct Sum decomposition of Zn [ω]J.(Theorem 2.2.3, Theorem 2.2.4), these results are very useful for our following studies.
In the third chapter, on the basis of the second chapter, we discuss the diameter (Theorem 3.1.2), planarity (Theorem 3.2.4) and girth (Theorem 3.3.2) of Z n[ω]. The major findings of this chapter published in the Journal of Hunan University of Technology in 2009 (2).
In the fourth chapter,we mainly discuss the undirected zero divisor graphs of a kind of non- commutative rings(the formal triangular matrix rings). In many materials and thesis of ring theory, they appear frequently as counter-examples. In the first section of this chapter, when M is a torsionless R ?module, we have given the set of zero divisors of an arbitrary formal triangular matrix ring.In the second section of this chapter, we make some discussions about the diameter of the zero divisor graphs of the formal triangular matrix rings (theorem4.2.4, theorem4.2.6, theorem 4.2.9).Theorem 4.2.4 points out that the diameters of the zero divisor graph of an arbitrary formal triangular matrix ring are 2 or 3. Theorem 4.2.6 provides a necessary condition for that the diameter equals 2. And then theorem 4.2.9 gives a necessary and sufficient condition for that the diameter equals 2 when a given condition is satisfied. These three ones are of great values to studies on the zero divisor graphs of formal triangular matrix rings. In the final section of this chapter, we discuss the diameter of the undirected zero divisor graphs of 2×2 upper triangular matrix rings (proposition 4.3.2- propositions 4.3.6), and then, we obtain the last theorem of this paper, that is, when the diameter of the undirected zero divisor graph of 2×2 upper triangular matrix rings on an arbitrary commutative ring is 3(Theorem 4.3.7).
The last part of the paper is the concluding remarks of this paper, on one hand, we summarize the main tasks and introduce the future studies of this paper; On the other hand, the author describes some questions on zero divisor graphs, which are very interesting to himself.
Z [i]和Z [ω]是抽象代数中环论的两个重要的环,常被作为例子散见于各类抽象代数教材及论文中,本文讨论了代数整数环Z [ω]的模n剩余类环Z n[ω]的零因子集合、单位乘群、素谱以及Zn [ω]J的局部环直和分解,并在这些结果的基础上,研究了Z n[ω]的零因子图的直径、平面性及围长;本文还讨论了形式三角矩阵环的无向零因子图的直径,并简要地研究了任意含幺交换环上二阶上三角矩阵环的零因子图.具体地说,全文共分为四章:第一章介绍了环的零因子图的发展历史,本文的研究背景,理论来源和研究意义.并给出了本文所用到的一些基本概念与符号.
第二章主要讨论了代数整数环Z [ω]的模n剩余类环Z n[ω]的零因子集合、单位乘群、素谱以及Zn [ω]J的局部环直和分解(定理2.2.3,定理2.2.4),这些结果对我们后面的研究是非常有用的.
第三章主要是探讨有限含幺交换环的零因子图的若干图论性质,从而直观的反映环的零因子的内部结构.在第二章的基础上,讨论了有限交换环Z n[ω]的零因子图的直径(定理3.1.2)、平面性(定理3.2.4)与围长(定理3.3.2),本章的主要结果即将在湖南工业大学学报2009年第2期发表.
第四章主要讨论一类非交换环即形式三角矩阵环的无向零因子图.形式三角矩阵环是一类非常重要的非交换环,在环论的许多教材及文献中,它经常被作为反例出现.本章第一节中,在M为R无扰模的条件下,我们给出了形式三角矩阵环的零因子集合;本章第二节,我们对形式三角矩阵环的无向零因子图的直径做了一些探讨(定理4.2.4,定理4.2.6,定理4.2.9),定理4.2.4证明了任意形式三角矩阵环的零因子图直径只能为2或3.紧接着,定理4.2.6在M为R无扰模的条件下,给出了任意形式三角矩阵环的零因子图直径为2的一个必要条件,而定理4.2.9则在某一前提下,为我们提供了形式三角矩阵环的无向零因子图直径为2的一个充要条件,这三个定理对于形式三角矩阵环的零因子图的研究是非常有意义的.在这一章的最后一节中,我们利用本章前两节的结论讨论了任意含幺交换环上二阶上三角矩阵环的无向零因子图的直径(命题4.3.2-命题4.3.6),得到了本文的最后一个定理,即任意含幺交换环上二阶上三角矩阵环的无向零因子图的直径为3的一个判定定理(定理4.3.7).
文章的最后部分是本文的结束语,一方面,我们总结了本文的主要工作并介绍了本文的若干后续工作;另一方面,我们阐述了作者关于环的零因子图的几个十分感兴趣的问题.
Ring Theory and Graph Theory are two very important mathematical branches, which are not only of great theoretical interest in themselves but also found important applications in many other branches of math (such as combinatorial mathematics, geometry, automata theory and coding theory, etc.). The zero divisor graphs of rings, using properties of graphs to study algebraic structures, has become an exciting research topic in the last twenty years, leading to many fascinating results and questions. In the past ten years, it has become a hot research field.
Z [i] and Z [ω] are two important rings in Abstract Algebra, often as scattered examples in materials and thesis of various types of abstract algebra. In this paper, at first, we discuss the set of zero divisors, units group, prime spectrum of Z n[ω], and the local ring Direct Sum decomposition of Zn [ω]J. On the basis of these results, we give the diameter, the girth and planarity of the zero divisor graphs of Z n[ω]. In this paper, we also study the zero divisor graphs of formal triangular matrix rings, and briefly study the zero divisor graphs of 2×2 upper triangular matrix rings. Specifically, the text is divided into four chapters:
In the first chapter, we summarize the history of the zero-divisor graph, and the background of this paper. At the same time, we give some notations and basic definitions of ring theory and graph theory.
In the second chapter, we discuss the set of zero-divisors , units group, prime spectrum of Z n[ω] and the local ring Direct Sum decomposition of Zn [ω]J.(Theorem 2.2.3, Theorem 2.2.4), these results are very useful for our following studies.
In the third chapter, on the basis of the second chapter, we discuss the diameter (Theorem 3.1.2), planarity (Theorem 3.2.4) and girth (Theorem 3.3.2) of Z n[ω]. The major findings of this chapter published in the Journal of Hunan University of Technology in 2009 (2).
In the fourth chapter,we mainly discuss the undirected zero divisor graphs of a kind of non- commutative rings(the formal triangular matrix rings). In many materials and thesis of ring theory, they appear frequently as counter-examples. In the first section of this chapter, when M is a torsionless R ?module, we have given the set of zero divisors of an arbitrary formal triangular matrix ring.In the second section of this chapter, we make some discussions about the diameter of the zero divisor graphs of the formal triangular matrix rings (theorem4.2.4, theorem4.2.6, theorem 4.2.9).Theorem 4.2.4 points out that the diameters of the zero divisor graph of an arbitrary formal triangular matrix ring are 2 or 3. Theorem 4.2.6 provides a necessary condition for that the diameter equals 2. And then theorem 4.2.9 gives a necessary and sufficient condition for that the diameter equals 2 when a given condition is satisfied. These three ones are of great values to studies on the zero divisor graphs of formal triangular matrix rings. In the final section of this chapter, we discuss the diameter of the undirected zero divisor graphs of 2×2 upper triangular matrix rings (proposition 4.3.2- propositions 4.3.6), and then, we obtain the last theorem of this paper, that is, when the diameter of the undirected zero divisor graph of 2×2 upper triangular matrix rings on an arbitrary commutative ring is 3(Theorem 4.3.7).
The last part of the paper is the concluding remarks of this paper, on one hand, we summarize the main tasks and introduce the future studies of this paper; On the other hand, the author describes some questions on zero divisor graphs, which are very interesting to himself.
引文
[1] I. Beck. Coloring of commutative rings[J]. J. Algebra, 1988,116 : 208–226.
[2] D. D. Anderson, M. Naseer.Beck’s coloring of a commutative ring[J]. J. Algebra,1993, 159 :500-514.
[3] D.F.Anderson and P.S. Livingston.The zero-divisor graph of a commutative ring[J]. J. Algebra,1999,217 : 434–447.
[4] F. DeMeyer, K. Schneider. Automorphisms and zero-divisor graphs of commutative rings[J]. Internat. J. Commutative Rings ,2002,1 (3):93–106.
[5] S. B. Mulay.Cycles and symmetries of zero-divisors[J]. Comm.Algebra,2002,30(7) : 3533– 3558.
[6] M.Axtell, J.Coykendall, J.Stickles. Zero-divisor graphs of polynomials and power series over commutative rings[J].Comm. Algebra, 2005, 33(6):2043-2050.
[7] D.F. Anderson, R. Levy, J. Shapiro. Zero-divisor graphs, von Neumann regular rings, and Boolean algebras[J].J. Pure Appl. Algebra,2003,180:221–241.
[8] M. Axtel, J. Stickles.Zero-divisor graphs of idealizations[J]. J. Pure Appl. Algebra,2006, 204:235–243.
[9]唐高华,苏华东,赵寿祥. [ ]Z ni的零因子图性质[J].广西师范大学学报(自科版), 2007, 25(3):32-35.
[10] S. Akbari, H. R. Maimani and S. Yassemi. When a zero-divisor graph is planar or a complete r-partite graph [J].J. Algebra, 2003,270 (1) :169–180.
[11] R. Belshoff, J. Chapman. Planar zero-divisor graphs[J]. J. Algebra, 2007,316: 471–480.
[12] D.C.Lu, T.S.Wu. On endomorphism-regularity of zero-divisor graphs[J]. Discrete Math, 2008,308:4811-4815.
[13] S. P. Redmond.An ideal-based zero-divisor graph of a commutative ring[J].Comm. Algebra,2003,31(9):4425-4442.
[14] S. P. Redmond.The zero-divisor graph of a non-commutative ring[J].Internat. J.Commu- tative Rings,2002,1(4): 203-211.
[15] S. Akbari, A. Mohammadian.Zero-divisor graphs of non-commutative rings[J]. J. Algebra, 2006,296(2):462-479.
[16] T. S. Wu.On directed zero-divisor graphs of finite rings[J]. Discrete Math, 2005,296(1): 73-86.
[17] S. P. Redmond. Central sets and radius of the zero-divisor graphs of commutative rings[J]. Comm. Algebra, 2006,34:2389-2401.
[18] F. DeMeyer, T. McKenzie, K. Schneider. The zero-divisor graph of a commutativesemigroup [J]. Semigroup Forum, 2002,65(2):206-214.
[19] F. DeMeyer, L. DeMeyer. Zero divisor graphs of semigroups[J]. J. Algebra, 2005,283(1): 190-198.
[20] T. S.Wu, F. Cheng.The structures of zero-divisor semigroups with graph Kn K2[J]. Semi-group Forum,2006,76:330-340.
[21] M. Zuo,T. S.Wu. A new graph structure of commutative semigroups[J]. Semi-group Forum,2005,70:71-80.
[22] T. S. Wu, D. C. Lu. Zero-divisor semigroups and some simple graphs[J].Comm. Algebra, 2006, 34(8): 3043-3052.
[23] K.R. Goodearl. Ring Theory, Nonsingular Rings and Modules[M].Marcel Dekker, Inc. New York and Basel,1976.
[24] A. Haghany and K. Varadarajan. Study of formal triangular matrix rings[J]. Comm. Algebra,1999,27(11):5507-5525.
[25]陈焕艮.关于形式三角矩阵环[J].南京大学数学半年刊,1999,16(2):153-157.
[26]王朝瑞.图论(第三版) [M].北京:北京理工大学出版社, 2004.
[27]潘承洞,潘承彪.初等数论[M] .北京:北京大学出版社,2005.
[28] Frank W. Anderson, Kent R. Fuller. Rings and Categories of Modules[M]. New York: Springer- Verlag , 1992.
[29]于萍,欧晓斌.代数整数环Z [ω]的素元及剩余类环[J].西安文理学院学报(自科版),2007,10(3):111-113.
[30]苏华东,唐高华. [ ]Z ni的素谱和零因子[J].广西师范学院学报(自科版), 2006, 23(4): 1-4.
[31]宋光天.交换代数导引[M].合肥:中国科学技术大学出版社,2002.
[32]冯克勤.交换代数基础[M] .北京:高等教育出版社,1985.
[33]杨子胥.近世代数[M].北京:高等教育出版社,2003.
[34] Jacobson N. Basic Algebra[M] . New York : W H Freeman and Co , 1980.
[35]王芳贵.交换环与星型算子理论[M].北京:科技出版社,2006.
[36] M. Axtell, J. Stickles, and J. Warfel. Zero-Divisor Graphs of Direct Products of Commutative Rings[J].Houston J.Math,2006,32(4):985-993.
[37] Thomas G .Lucas. The diameter of zero divisor graph[J].J.Algebra,2006,301:174-193.
[2] D. D. Anderson, M. Naseer.Beck’s coloring of a commutative ring[J]. J. Algebra,1993, 159 :500-514.
[3] D.F.Anderson and P.S. Livingston.The zero-divisor graph of a commutative ring[J]. J. Algebra,1999,217 : 434–447.
[4] F. DeMeyer, K. Schneider. Automorphisms and zero-divisor graphs of commutative rings[J]. Internat. J. Commutative Rings ,2002,1 (3):93–106.
[5] S. B. Mulay.Cycles and symmetries of zero-divisors[J]. Comm.Algebra,2002,30(7) : 3533– 3558.
[6] M.Axtell, J.Coykendall, J.Stickles. Zero-divisor graphs of polynomials and power series over commutative rings[J].Comm. Algebra, 2005, 33(6):2043-2050.
[7] D.F. Anderson, R. Levy, J. Shapiro. Zero-divisor graphs, von Neumann regular rings, and Boolean algebras[J].J. Pure Appl. Algebra,2003,180:221–241.
[8] M. Axtel, J. Stickles.Zero-divisor graphs of idealizations[J]. J. Pure Appl. Algebra,2006, 204:235–243.
[9]唐高华,苏华东,赵寿祥. [ ]Z ni的零因子图性质[J].广西师范大学学报(自科版), 2007, 25(3):32-35.
[10] S. Akbari, H. R. Maimani and S. Yassemi. When a zero-divisor graph is planar or a complete r-partite graph [J].J. Algebra, 2003,270 (1) :169–180.
[11] R. Belshoff, J. Chapman. Planar zero-divisor graphs[J]. J. Algebra, 2007,316: 471–480.
[12] D.C.Lu, T.S.Wu. On endomorphism-regularity of zero-divisor graphs[J]. Discrete Math, 2008,308:4811-4815.
[13] S. P. Redmond.An ideal-based zero-divisor graph of a commutative ring[J].Comm. Algebra,2003,31(9):4425-4442.
[14] S. P. Redmond.The zero-divisor graph of a non-commutative ring[J].Internat. J.Commu- tative Rings,2002,1(4): 203-211.
[15] S. Akbari, A. Mohammadian.Zero-divisor graphs of non-commutative rings[J]. J. Algebra, 2006,296(2):462-479.
[16] T. S. Wu.On directed zero-divisor graphs of finite rings[J]. Discrete Math, 2005,296(1): 73-86.
[17] S. P. Redmond. Central sets and radius of the zero-divisor graphs of commutative rings[J]. Comm. Algebra, 2006,34:2389-2401.
[18] F. DeMeyer, T. McKenzie, K. Schneider. The zero-divisor graph of a commutativesemigroup [J]. Semigroup Forum, 2002,65(2):206-214.
[19] F. DeMeyer, L. DeMeyer. Zero divisor graphs of semigroups[J]. J. Algebra, 2005,283(1): 190-198.
[20] T. S.Wu, F. Cheng.The structures of zero-divisor semigroups with graph Kn K2[J]. Semi-group Forum,2006,76:330-340.
[21] M. Zuo,T. S.Wu. A new graph structure of commutative semigroups[J]. Semi-group Forum,2005,70:71-80.
[22] T. S. Wu, D. C. Lu. Zero-divisor semigroups and some simple graphs[J].Comm. Algebra, 2006, 34(8): 3043-3052.
[23] K.R. Goodearl. Ring Theory, Nonsingular Rings and Modules[M].Marcel Dekker, Inc. New York and Basel,1976.
[24] A. Haghany and K. Varadarajan. Study of formal triangular matrix rings[J]. Comm. Algebra,1999,27(11):5507-5525.
[25]陈焕艮.关于形式三角矩阵环[J].南京大学数学半年刊,1999,16(2):153-157.
[26]王朝瑞.图论(第三版) [M].北京:北京理工大学出版社, 2004.
[27]潘承洞,潘承彪.初等数论[M] .北京:北京大学出版社,2005.
[28] Frank W. Anderson, Kent R. Fuller. Rings and Categories of Modules[M]. New York: Springer- Verlag , 1992.
[29]于萍,欧晓斌.代数整数环Z [ω]的素元及剩余类环[J].西安文理学院学报(自科版),2007,10(3):111-113.
[30]苏华东,唐高华. [ ]Z ni的素谱和零因子[J].广西师范学院学报(自科版), 2006, 23(4): 1-4.
[31]宋光天.交换代数导引[M].合肥:中国科学技术大学出版社,2002.
[32]冯克勤.交换代数基础[M] .北京:高等教育出版社,1985.
[33]杨子胥.近世代数[M].北京:高等教育出版社,2003.
[34] Jacobson N. Basic Algebra[M] . New York : W H Freeman and Co , 1980.
[35]王芳贵.交换环与星型算子理论[M].北京:科技出版社,2006.
[36] M. Axtell, J. Stickles, and J. Warfel. Zero-Divisor Graphs of Direct Products of Commutative Rings[J].Houston J.Math,2006,32(4):985-993.
[37] Thomas G .Lucas. The diameter of zero divisor graph[J].J.Algebra,2006,301:174-193.