二面体群上的群环的零因子图
摘要
群环是一个重要的环类,它不仅与群论,环论有关,而且与域论,线性代数,代数数论,代数拓扑等理论具有紧密的联系.近年来,群环在密码,通信等领域都有着广泛的应用.使用图性质研究代数系统,已成为近年来数学研究的一个热点问题,引发了许多有趣的结果和问题.本文主要研究了非交换群环ZnDm的代数性质,结构,以及零因子图的性质,其中Zn为模n剩余类环,Dm为2m阶二面体群.
本文分为六个部分,第一部分为引言,第二至第五部分中每一部分为一章,是本文的核心,最后部分是结束语.
在本文的第一章中我们概述了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.同时我们还给出了环论和图论的一些基本概念与结论.
在第二章中我们对非交换群环ZnDm的代数性质及其结构进行了研究,特别是对当m=p和m=2t时,非交换群环ZnDm的代数结构,给出了较为具体的刻画(定理2.2.1,命题2.2.1,推论2.2.1,推论2.2.2,定理2.3.1,推论2.3.1).
在第三章中,我们主要讨论了非交换群环ZnDm的有向零因子图的围长；直径；平面性,分别给出了较为具体的刻画(定理3.3.1,定理3.2.1,定理3.2.2,定理3.2.3).特别地,对当m=2t+1,(t为正整数)时,非交换群环ZnDm的无向零因子图的中心给出了一个具体的刻画(定理3.3.1).
在第四章中,我们对非交换群环ZnD4的代数结构和零因子情况进行了详细的讨论(定理4.1.1,定理4.1.2,推论4.1.2,推论4.1.3),其中D4表示8阶二面体群,同时,对非交换群环ZnD4的零因子图性质进行了刻画(定理4.2.1,定理4.2.2,定理4.2.3,定理4.1.4).本章的主要结果近期将在《广西师范大学学报》(自然科学版)发表(已接受).
在第五章中,我们讨论了当群环ZnG为有限局部群环时,其代数结构和零因子情况(定理5.1.1,定理5.1.2),其中Zn为模n剩余类环,G为有限群.并讨论了群环ZnG的零因子图的围长；直径；平面性；中心等性质(定理5.2.1,定理5.2.2,定理5.2.3,定理5.2.4).本章的主要结果已于2010年10月发表在《广西民族师范学院学报》.
最后部分为结束语,总结了本文的主要工作,阐述了与本文相关研究的一些课题,并对下一步的继续研究工作做了设想.
Group rings are very important algebraic structures in ring theory. Besides the obvious relationship with group theory and ring theory, they are related to field theory, linear algebra, algebraic number theory, algebraic topology etc. Recently, group rings have been widely applied to the fields of codes and communications. The study of algebraic structures, using the properties of graphs, becomes an exciting research topic in the last twenty years, leading to many fascinating results and questions. This paper mainly discusses algebraic properties, structures, and properties of zero-divisor graphs with regard to the noncommutative group rings ZnDm, where Zn is the residue class ring modulo n and Dm is a dihedral group of order 2m.
This paper is composed of six parts, where the first part is the introduction, the second to the fifth parts in which each part is a chapter are the core of the paper, and the last part is the concluding remarks.
In Chapter 1 of this paper, we summarize the history of the zero-divisor graph, the research background, theory source and research significance of this paper. At the same time, we give the notation and basic results of ring theory and graph theory.
In Chapter 2, we investigate the algebraic properties of noncommutative group rings ZnDm and their structures. Specially, when m=p and m=2t, we give the detailed characterization to their algebraic structures [Theorem 2.2.1, Proposition 2.2.1, Corollary 2.2.1, Theorem 2.3.1, Proposition 2.3.1, Corollary 2.3.1].
In Chapter 3, we mainly discuss the girth, the diameter and the planarity of zero-divisor graph of ZnDm[Theorem 3.1.1, Theorem 3.2.1, Theorem 3.2.2, Theorem 3.2.3]. Specially, we give the center of the undirected zero-divisors graph, when m—2t+1,(t is a positive integer)[Theorem 3.3.1].
In Chapter 4, we mainly investigate the algebraic structures of group rings ZnD4 and their zero-divisors. At the same time, we give the detailed characterizations to them re-spectively[Theorem4.1.1, Theorem 4.1.2, Corollary 4.1.2, Corollary 4.1.3], where D4 is a dihedral group of order 8. Also, we give the properties of zero-divisor graph of group rings ZnD4. The main results of this section have been accepted and will be published in Journal of Guangxi Normal University.
In Chapter 5, we mainly discuss the algebraic structures and zero-divisors of group ring ZnG, when ZnG is a finite local group ring[Theorem5.1.1, Theorem 5.1.2]. At the same time we discuss the girth, the diameter, the planarity and center of zero-divisor graph of ZnG[Theorem 5.2.1, Theorem 5.2.2, Theorem 5.2.3, Theorem 5.2.4]. The main results of this section have been published in Journal of Guangxi Normal University for Nationalities.
The last part is the concluding remarks, the main work of this paper is summarized, some subjects corresponding to this paper are simply illustrated, and the idea to the next phase of work to continue to study is roughly envisaged.
本文分为六个部分,第一部分为引言,第二至第五部分中每一部分为一章,是本文的核心,最后部分是结束语.
在本文的第一章中我们概述了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.同时我们还给出了环论和图论的一些基本概念与结论.
在第二章中我们对非交换群环ZnDm的代数性质及其结构进行了研究,特别是对当m=p和m=2t时,非交换群环ZnDm的代数结构,给出了较为具体的刻画(定理2.2.1,命题2.2.1,推论2.2.1,推论2.2.2,定理2.3.1,推论2.3.1).
在第三章中,我们主要讨论了非交换群环ZnDm的有向零因子图的围长；直径；平面性,分别给出了较为具体的刻画(定理3.3.1,定理3.2.1,定理3.2.2,定理3.2.3).特别地,对当m=2t+1,(t为正整数)时,非交换群环ZnDm的无向零因子图的中心给出了一个具体的刻画(定理3.3.1).
在第四章中,我们对非交换群环ZnD4的代数结构和零因子情况进行了详细的讨论(定理4.1.1,定理4.1.2,推论4.1.2,推论4.1.3),其中D4表示8阶二面体群,同时,对非交换群环ZnD4的零因子图性质进行了刻画(定理4.2.1,定理4.2.2,定理4.2.3,定理4.1.4).本章的主要结果近期将在《广西师范大学学报》(自然科学版)发表(已接受).
在第五章中,我们讨论了当群环ZnG为有限局部群环时,其代数结构和零因子情况(定理5.1.1,定理5.1.2),其中Zn为模n剩余类环,G为有限群.并讨论了群环ZnG的零因子图的围长；直径；平面性；中心等性质(定理5.2.1,定理5.2.2,定理5.2.3,定理5.2.4).本章的主要结果已于2010年10月发表在《广西民族师范学院学报》.
最后部分为结束语,总结了本文的主要工作,阐述了与本文相关研究的一些课题,并对下一步的继续研究工作做了设想.
Group rings are very important algebraic structures in ring theory. Besides the obvious relationship with group theory and ring theory, they are related to field theory, linear algebra, algebraic number theory, algebraic topology etc. Recently, group rings have been widely applied to the fields of codes and communications. The study of algebraic structures, using the properties of graphs, becomes an exciting research topic in the last twenty years, leading to many fascinating results and questions. This paper mainly discusses algebraic properties, structures, and properties of zero-divisor graphs with regard to the noncommutative group rings ZnDm, where Zn is the residue class ring modulo n and Dm is a dihedral group of order 2m.
This paper is composed of six parts, where the first part is the introduction, the second to the fifth parts in which each part is a chapter are the core of the paper, and the last part is the concluding remarks.
In Chapter 1 of this paper, we summarize the history of the zero-divisor graph, the research background, theory source and research significance of this paper. At the same time, we give the notation and basic results of ring theory and graph theory.
In Chapter 2, we investigate the algebraic properties of noncommutative group rings ZnDm and their structures. Specially, when m=p and m=2t, we give the detailed characterization to their algebraic structures [Theorem 2.2.1, Proposition 2.2.1, Corollary 2.2.1, Theorem 2.3.1, Proposition 2.3.1, Corollary 2.3.1].
In Chapter 3, we mainly discuss the girth, the diameter and the planarity of zero-divisor graph of ZnDm[Theorem 3.1.1, Theorem 3.2.1, Theorem 3.2.2, Theorem 3.2.3]. Specially, we give the center of the undirected zero-divisors graph, when m—2t+1,(t is a positive integer)[Theorem 3.3.1].
In Chapter 4, we mainly investigate the algebraic structures of group rings ZnD4 and their zero-divisors. At the same time, we give the detailed characterizations to them re-spectively[Theorem4.1.1, Theorem 4.1.2, Corollary 4.1.2, Corollary 4.1.3], where D4 is a dihedral group of order 8. Also, we give the properties of zero-divisor graph of group rings ZnD4. The main results of this section have been accepted and will be published in Journal of Guangxi Normal University.
In Chapter 5, we mainly discuss the algebraic structures and zero-divisors of group ring ZnG, when ZnG is a finite local group ring[Theorem5.1.1, Theorem 5.1.2]. At the same time we discuss the girth, the diameter, the planarity and center of zero-divisor graph of ZnG[Theorem 5.2.1, Theorem 5.2.2, Theorem 5.2.3, Theorem 5.2.4]. The main results of this section have been published in Journal of Guangxi Normal University for Nationalities.
The last part is the concluding remarks, the main work of this paper is summarized, some subjects corresponding to this paper are simply illustrated, and the idea to the next phase of work to continue to study is roughly envisaged.
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.Zn[i]的零因子图性质[J].广西师范大学学报(自然科学版),2007,25(3)：32-35.
[10]郭述锋.群环的零因子图.广西师范大学硕士论文,2009.
[11]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.
[12]R. Belshoff, J. Chapman, Planar zero-divisor graphs [J], J. Algebra,2007,316:471-480.
[13]R. Akhtar, L. Lee, Connectivity of the zero-divisor graph for finite rings[J], preprint 2005.
[14]K. Samei, The zero-divisor graph of a reduced ring[J], J. Pure Appl. Algebra,2007,209: 813-821.
[15]S. P. Redmond, The zero-divisor graph of a non-commutative ring[J], Internat. J. Commu-tative Rings,2002,1(4):203-211.
[16]S. Akbari, A. Mohammadian,Zero-divisor graphs of non-commutative rings[J], J. Algebra, 2006,296(2):462-479.
[17]T. S. Wu, On directed zero-divisor graphs of finite rings [J], Discrete Math.,2005,296(1): 73-86.
[18]C. P. Milies, S. K. Sehgal, An introduction to Group Rings[M], Kluwer Acadimic Publishers, Dordrecht,2002.
[19]D.S.Passman, The algebraic structure of group rings,Wiley-interscience,New York,1977.
[20]T.Y.Lam,A First Course in Noncommutative Rings,Springer,New Yerk,2001.
[21]G.H.Hardy,E.M.Wright(著),张明尧,张凡(译).数论导引[M]北京：人民邮电出版社.
[22]I.G.Connell.on the group ring. Canad.J.Math.15,650-685;(1963).
[23]Juncheol Han.The Zero-Divisor Graph under Group Actions in a Noncommutative Ring [J],Journal of the Korean Mathematical Society,2008,45(6):1647-1659.
[24]F. Buckley, M. Lewinter(著).李慧霸,王风芹(译).图论简明教程[M].北京：清华大学出版社,2005.
[25]Nicholson W K.Local group rings [J].Canad Math Bull,1972,15(1):137-138.
[26]郭述锋,徐承杰,易忠.群环ZnG的代数性质及其结构[J].广西师范大学学报：自然科学版,2009,27(2)：42-45.
[27]Yuji Kobayashi,Kwangil Koh,A classification of finite rings by zero divisors,Journal of Pure and Applied Algebra,40(1986)135-147.
[28]潘承洞,潘承彪.初等数论[M].北京：北京大学出版社,2005.
[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.Zn[i]的零因子图性质[J].广西师范大学学报(自然科学版),2007,25(3)：32-35.
[10]郭述锋.群环的零因子图.广西师范大学硕士论文,2009.
[11]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.
[12]R. Belshoff, J. Chapman, Planar zero-divisor graphs [J], J. Algebra,2007,316:471-480.
[13]R. Akhtar, L. Lee, Connectivity of the zero-divisor graph for finite rings[J], preprint 2005.
[14]K. Samei, The zero-divisor graph of a reduced ring[J], J. Pure Appl. Algebra,2007,209: 813-821.
[15]S. P. Redmond, The zero-divisor graph of a non-commutative ring[J], Internat. J. Commu-tative Rings,2002,1(4):203-211.
[16]S. Akbari, A. Mohammadian,Zero-divisor graphs of non-commutative rings[J], J. Algebra, 2006,296(2):462-479.
[17]T. S. Wu, On directed zero-divisor graphs of finite rings [J], Discrete Math.,2005,296(1): 73-86.
[18]C. P. Milies, S. K. Sehgal, An introduction to Group Rings[M], Kluwer Acadimic Publishers, Dordrecht,2002.
[19]D.S.Passman, The algebraic structure of group rings,Wiley-interscience,New York,1977.
[20]T.Y.Lam,A First Course in Noncommutative Rings,Springer,New Yerk,2001.
[21]G.H.Hardy,E.M.Wright(著),张明尧,张凡(译).数论导引[M]北京：人民邮电出版社.
[22]I.G.Connell.on the group ring. Canad.J.Math.15,650-685;(1963).
[23]Juncheol Han.The Zero-Divisor Graph under Group Actions in a Noncommutative Ring [J],Journal of the Korean Mathematical Society,2008,45(6):1647-1659.
[24]F. Buckley, M. Lewinter(著).李慧霸,王风芹(译).图论简明教程[M].北京：清华大学出版社,2005.
[25]Nicholson W K.Local group rings [J].Canad Math Bull,1972,15(1):137-138.
[26]郭述锋,徐承杰,易忠.群环ZnG的代数性质及其结构[J].广西师范大学学报：自然科学版,2009,27(2)：42-45.
[27]Yuji Kobayashi,Kwangil Koh,A classification of finite rings by zero divisors,Journal of Pure and Applied Algebra,40(1986)135-147.
[28]潘承洞,潘承彪.初等数论[M].北京：北京大学出版社,2005.