三次对称群上的群环的零因子图
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摘要
群环的零因子图的研究涉及到数学中许多领域：环论,群论,半群论,域论,图论和初等数论等.如此众多的学科交叉在一起,使它不但具有吸引力和趣味性,而且也有许多内在的困难.使用图的性质研究代数系统,已成为近20年来数学研究的一个热点问题,引发了很多有趣的结果和问题.
本文分为五个部分,第一部分为引言,第二至第四部分各为一章,是本文的核心,最后部分是小结以及进一步研究的问题.
第一章中我们概述了零因子图发展的历史,研究现状以及本文的研究背景,同时我们还给出了环论和图论的一些基本的概念.
第二章中我们研究了群环ZnS3的零因子,并对ZnS3的有向零因子图Γ(ZnS3)的直径、围长进行了较为具体的刻画.这些结果对我们后面的研究是非常有用的.本章的主要结果已经在《广西师范大学学报》12(4)(2010)上发表.主要结果如下：
定理2.3设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,则R的零因子为
定理2.4设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则
(1)diam(Γ(R))=2(?)n=3t,t≥1;
(2)diam(Γ(R))=3(?)n≠3t,t≥1.
定理2.5设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则gr(Γ(ZnS3))=3.
定理2.6设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则Γ(ZnS3)是非平面图.
第三章在第二章的基础上,将环推广到任意的非零有限交换环R上,我们对群环RS3的代数性质及其结构进行了研究,给出了较为具体的刻画,并对RS3的零因子图的直径、围长、平面性进行了刻画.主要结果有：
定理3.4设R为特征为pr的有限环,S3是三次对称群,若(p,2)=(p,3)=1,则diam(Γ(RS3))=3.
定理3.6设R为有限环,S3为三次对称群,则qr(Γ(RS3))=3.
定理3.7设R为有限环,S3为三次对称群,则Γ(RS3)为非平面图.
第四章将三次对称群推广到m次对称群,当p(?)m!时,对ZpSm做一些有意义的探索,初步刻画了ZpSm的代数结构,并对其围长及平面性进行了具体的刻画.
定理4.2设Zp为模p剩余类环,Sm为m次对称群,当p(?)m!时,有ZpSm≌Zp(?)Zp(?)△(Sm,Am)
定理4.4设Zn为模n剩余类环,Am为m次交错群,若m≥4,则gr(Γ(ZnAm))=3.
定理4.5设Zn为模n剩余类环,Sm为m次对称群,若m≥3,则gr(Γ(ZnSm))=3.
定理4.7设Zn为模n剩余类环,Am为m次交错群,若m≥4,则Γ(ZnAm))为非平面图.
定理4.8设Zn为模n剩余类环,Sm为m次对称群,若m≥3,则Γ(ZnSm)为非平面图.
The study of the zero-divisor graphs are related to many areas:ring theory,group theory, semigroup theory,field theory,graph theory and elementary number theory. So many subjects intercross,make it fascinated and attracted,but also inherent dilemmas.The study of algebraic structures,using properties of graphs,has become an exciting research topic in the last twenty years,leading to many fascinating results and questions.
This paper is composed of five parts,where the first part is the introduction,the second to the forth in which each part is a chapter are the core of the paper,and the last part is the further research problems.
In Chapter 1 of this paper,we summarize the history and current situation of the zero-divisor graph,the research background of this paper.At the same time,we gave some notations of ring theory and graph theory.
In Chapter 2,we will determine the zero-divisors of ZnS3,and discuss the girth and diameter of the directed zero-divisor graphΓ(ZnS3).The main result has been published in Journal of Guangxi Normal University 12(4)(2010).These results will be very useful in the following chapters,and they are as following:
Theorem 2.3 Let Zn be the modulo n residue class ring,S3 be the symmetric group on three letters,R=ZnS3,then the zero-divisors of R are
(1)If n=2,then D(R)={α∈R││supp(α)│=0,2,3,4,6},│D(R)│=52;
(2)If n=3,then D(R)={α=x1+x2b+x3ab+x4a2b+x5a+x6a2│xi∈Z3,x1+x5+x6≡x2+x3+x4(mod 3),or (?)xi≡0(mod 3)},│D(R)│=405;
(3)If n=p,p>3 is a prime number,then ZpS3≌Zp⊕Zp⊕M2(Zp),and D(Zp⊕Zp⊕M2(Zp))-{(α1,α2(?))│at least one ofα1 andα│D(ZpS3)│=3p5-2p4-2p3+3p2-p;
(4)If n=pt,t>1,then ZptS3/IS3≌ZpS3.D(ZptS3=D(ZpS3)+IS3,in which I={0,p,2p,…,(pt-1-1)p}is the unique maximal ideal of Zpt;
(5)If n=p1r1p2r2…p3r3,ri≥1,s≥2,then R≌Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3, D(Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3)={(α1,α2,…,αs)│αi∈ZpiriS3,at least one ofαi∈D(ZpiriS3),i=1,2,3,…,s}.
Theorem 2.4 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3,R=ZnS3,n>1, then
(1) diam{Γ(R))=2(?)n=3t,t≥1;
(2) diam(Γ(R))=3(?)n≠3t,t≥1.
Theorem 2.5 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, then gr(Γ(ZnS3))=3.
Theorem 2.6 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, thenΓ(ZnS3) is non-planar.
In Chapter 3, we generalize the residue class ring to arbitrary finite ring R, we determine the algebraic proposition and structure of RS3, and discuss the girth and diameter of the directed zero-divisor graphΓ(RS3). The main results are:
Theorem 3.4 Let R be a finite ring of characteristic pT, S3 be the symmetric group on three letters, if (p,2)=(p,3)=1, then diam(Γ(RS3))=3.
Theorem 3.6 Let R be a finite ring, S3 be the symmetric group on three letters, then gr(r(RS3))=3.
Theorem 3.7 Let R be a finite ring, S3 be the symmetric group on three letters, thenΓ(RS3) is non-planar.
In Chapter 4, we generalize the symmetric group on three letters to m letters, when p│m!, we make some meaningful exploration of ZpSm, and discuss the girth and planarity. The main results are:
Theorem 4.2 Let Zp be the modulo p residue class ring, Sm be the symmetric group on m letters, if p│m!, then ZpSm≌Zp⊕Zp⊕△(Sm, Am)
Theorem 4.4 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, then gr(Γ(ZnAm))=3.
Theorem 4.5 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, then gr(Γ(ZnSm))=3.
Theorem 4.7 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, thenΓ(ZnAm)) is non-planar.
Theorem 4.8 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, thenΓ(ZnSm) is non-planar.
本文分为五个部分,第一部分为引言,第二至第四部分各为一章,是本文的核心,最后部分是小结以及进一步研究的问题.
第一章中我们概述了零因子图发展的历史,研究现状以及本文的研究背景,同时我们还给出了环论和图论的一些基本的概念.
第二章中我们研究了群环ZnS3的零因子,并对ZnS3的有向零因子图Γ(ZnS3)的直径、围长进行了较为具体的刻画.这些结果对我们后面的研究是非常有用的.本章的主要结果已经在《广西师范大学学报》12(4)(2010)上发表.主要结果如下：
定理2.3设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,则R的零因子为
定理2.4设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则
(1)diam(Γ(R))=2(?)n=3t,t≥1;
(2)diam(Γ(R))=3(?)n≠3t,t≥1.
定理2.5设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则gr(Γ(ZnS3))=3.
定理2.6设Zn为模n剩余类环,S3为3次对称群,R=ZnS3,n>1,则Γ(ZnS3)是非平面图.
第三章在第二章的基础上,将环推广到任意的非零有限交换环R上,我们对群环RS3的代数性质及其结构进行了研究,给出了较为具体的刻画,并对RS3的零因子图的直径、围长、平面性进行了刻画.主要结果有：
定理3.4设R为特征为pr的有限环,S3是三次对称群,若(p,2)=(p,3)=1,则diam(Γ(RS3))=3.
定理3.6设R为有限环,S3为三次对称群,则qr(Γ(RS3))=3.
定理3.7设R为有限环,S3为三次对称群,则Γ(RS3)为非平面图.
第四章将三次对称群推广到m次对称群,当p(?)m!时,对ZpSm做一些有意义的探索,初步刻画了ZpSm的代数结构,并对其围长及平面性进行了具体的刻画.
定理4.2设Zp为模p剩余类环,Sm为m次对称群,当p(?)m!时,有ZpSm≌Zp(?)Zp(?)△(Sm,Am)
定理4.4设Zn为模n剩余类环,Am为m次交错群,若m≥4,则gr(Γ(ZnAm))=3.
定理4.5设Zn为模n剩余类环,Sm为m次对称群,若m≥3,则gr(Γ(ZnSm))=3.
定理4.7设Zn为模n剩余类环,Am为m次交错群,若m≥4,则Γ(ZnAm))为非平面图.
定理4.8设Zn为模n剩余类环,Sm为m次对称群,若m≥3,则Γ(ZnSm)为非平面图.
The study of the zero-divisor graphs are related to many areas:ring theory,group theory, semigroup theory,field theory,graph theory and elementary number theory. So many subjects intercross,make it fascinated and attracted,but also inherent dilemmas.The study of algebraic structures,using properties of graphs,has become an exciting research topic in the last twenty years,leading to many fascinating results and questions.
This paper is composed of five parts,where the first part is the introduction,the second to the forth in which each part is a chapter are the core of the paper,and the last part is the further research problems.
In Chapter 1 of this paper,we summarize the history and current situation of the zero-divisor graph,the research background of this paper.At the same time,we gave some notations of ring theory and graph theory.
In Chapter 2,we will determine the zero-divisors of ZnS3,and discuss the girth and diameter of the directed zero-divisor graphΓ(ZnS3).The main result has been published in Journal of Guangxi Normal University 12(4)(2010).These results will be very useful in the following chapters,and they are as following:
Theorem 2.3 Let Zn be the modulo n residue class ring,S3 be the symmetric group on three letters,R=ZnS3,then the zero-divisors of R are
(1)If n=2,then D(R)={α∈R││supp(α)│=0,2,3,4,6},│D(R)│=52;
(2)If n=3,then D(R)={α=x1+x2b+x3ab+x4a2b+x5a+x6a2│xi∈Z3,x1+x5+x6≡x2+x3+x4(mod 3),or (?)xi≡0(mod 3)},│D(R)│=405;
(3)If n=p,p>3 is a prime number,then ZpS3≌Zp⊕Zp⊕M2(Zp),and D(Zp⊕Zp⊕M2(Zp))-{(α1,α2(?))│at least one ofα1 andα│D(ZpS3)│=3p5-2p4-2p3+3p2-p;
(4)If n=pt,t>1,then ZptS3/IS3≌ZpS3.D(ZptS3=D(ZpS3)+IS3,in which I={0,p,2p,…,(pt-1-1)p}is the unique maximal ideal of Zpt;
(5)If n=p1r1p2r2…p3r3,ri≥1,s≥2,then R≌Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3, D(Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3)={(α1,α2,…,αs)│αi∈ZpiriS3,at least one ofαi∈D(ZpiriS3),i=1,2,3,…,s}.
Theorem 2.4 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3,R=ZnS3,n>1, then
(1) diam{Γ(R))=2(?)n=3t,t≥1;
(2) diam(Γ(R))=3(?)n≠3t,t≥1.
Theorem 2.5 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, then gr(Γ(ZnS3))=3.
Theorem 2.6 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, thenΓ(ZnS3) is non-planar.
In Chapter 3, we generalize the residue class ring to arbitrary finite ring R, we determine the algebraic proposition and structure of RS3, and discuss the girth and diameter of the directed zero-divisor graphΓ(RS3). The main results are:
Theorem 3.4 Let R be a finite ring of characteristic pT, S3 be the symmetric group on three letters, if (p,2)=(p,3)=1, then diam(Γ(RS3))=3.
Theorem 3.6 Let R be a finite ring, S3 be the symmetric group on three letters, then gr(r(RS3))=3.
Theorem 3.7 Let R be a finite ring, S3 be the symmetric group on three letters, thenΓ(RS3) is non-planar.
In Chapter 4, we generalize the symmetric group on three letters to m letters, when p│m!, we make some meaningful exploration of ZpSm, and discuss the girth and planarity. The main results are:
Theorem 4.2 Let Zp be the modulo p residue class ring, Sm be the symmetric group on m letters, if p│m!, then ZpSm≌Zp⊕Zp⊕△(Sm, Am)
Theorem 4.4 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, then gr(Γ(ZnAm))=3.
Theorem 4.5 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, then gr(Γ(ZnSm))=3.
Theorem 4.7 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, thenΓ(ZnAm)) is non-planar.
Theorem 4.8 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, thenΓ(ZnSm) is non-planar.
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]D. F. Anderson, R. Levy, J. Shapiro. Zero-divisor graphs, von Neumann regular rings, and Boolean algebra[J]. J. Pure Applied Algebra,2003,180(3):221-241.
[5]R. Levy, J. Shapiro. The zero-divisor graph of von Neumann regular rings[J]. Comm.Algebra, 2002,30(2):745-750.
[6]K. Samei, The zero-divisor graph of a reduced ring[J]. Journal of Pure and Applied Algebra, 2007,209,813-821.
[7]M. Axtel, J. Stickles. Zero-divisor graphs of idealizations [J]. J. Pure Appl, Algebra,2006, 204:235-243.
[8]M. Axtel, J. Coykendall, J. Stickles. Zero-divisor graphs of polynomials and power series over commutative rings[J]. Comm. Algebra,2005,33(6):2043-2050.
[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[D].广西：广西师范大学,2009.
[11]R. Akhtar, L. Lee. Connectivity of the zero-divisor graph for finite rings, preprint 2005.
[12]S. P. Redmond. Central sets and radii of the zero-divisor graphs of commutative rings[J]. Comm. Algebra,2006,34,2389-2401.
[13]S. Akbari, H. R. Maimani, S. Yassemi. When a zero-divisor graph is planar or a completer r-partite graph[J]. J. Algebar,2003,270(1):169-180.
[14]F. DeMeyer, K. Schneider, Automorphisms and zero-divisor graphs of commutative rings[J], Internat. J. Commutative Rings,2002,1(3):93-106.
[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]T. S. Wu. On directed zero-divisor graphs of finite rings[J]. Discrete Math,2005,296(1): 73-86.
[17]S. Akbari, A. Mohammadian. Zero-divisor graphs of non-commutative rings[J], J. Algebra, 2006,296(2):462-479.
[18]G. A. Cannon, K. M. Neuerburg, S. P. Redmond. Zero-divisor graph of near rings and semigourp, Nearring and Nearfielas[J]. Dordrecht:Springer,189-200.
[19]S. P. Redmond. Structure in the zero-divisor graph of a noncommutative ring[J]. Houston J. Math,2004,30(2),345-355.
[20]C. P. Milies, S. K. Sehgal, An introduction to Group Rings[M], Kluwer Acadimic Publishers, Dordrecht,2002:125-150.
[21]J.A.邦迪,U.S.R.然蒂(著).吴望名,李念组,吴兰芳,谢伟如,梁文沛(译).图论及其应用[M].北京：科学出版社,1984：161-165.
[22]潘承洞,潘承彪(著).初等数论(第二版)[M].北京：北京大学出版社,1992：129-130.
[23]杨子胥(著).近似代数(第二版)[M].北京：高等教育出版社,2003.
[24]Brace A.Magurn. An Algebraic Introduction to K-Theory[M]. Cambridge University Press, 2002:120-125.
[25]R. K. Sharma, J. B.S rivastava and Manju Kbian. The unit group of FA4[J]. Publ:Math Debrecen.2007,71:21-27.
[26]S. B. Mulay. Cycles and symmetries of zero-divisors [J]. Comm.Algebra,2002,30(7):3533-3558.
[27]R. Belshoff, J. Chapman, Planar zero-divisor graphs[J], J. Algebra,2007,316:471-480.
[28]D. S. Passman. The algebraic structure of group rings[M]. Wiley-interscience,New York,1977: 3-28.
[29]T. Y. Lam. A First Course in Noncommutative Rings[M]. New Yerk:Springer,2001:121-125.
[30]I. G. Connell. on the group ring[J]. Canad.J.Math.1963:15:650-685.
[31]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,2005：100-125.
[32]Nicholson W K. Local group rings [J].Canad Math Bull,1972,15(1):137-138.
[33]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.
[34]Yuji Kobayashi, Kwangil Koh. A classification of finite rings by zero divisors [J]. Journal of Pure and Applied Algebra,1986,40:135-147.
[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]D. F. Anderson, R. Levy, J. Shapiro. Zero-divisor graphs, von Neumann regular rings, and Boolean algebra[J]. J. Pure Applied Algebra,2003,180(3):221-241.
[5]R. Levy, J. Shapiro. The zero-divisor graph of von Neumann regular rings[J]. Comm.Algebra, 2002,30(2):745-750.
[6]K. Samei, The zero-divisor graph of a reduced ring[J]. Journal of Pure and Applied Algebra, 2007,209,813-821.
[7]M. Axtel, J. Stickles. Zero-divisor graphs of idealizations [J]. J. Pure Appl, Algebra,2006, 204:235-243.
[8]M. Axtel, J. Coykendall, J. Stickles. Zero-divisor graphs of polynomials and power series over commutative rings[J]. Comm. Algebra,2005,33(6):2043-2050.
[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[D].广西：广西师范大学,2009.
[11]R. Akhtar, L. Lee. Connectivity of the zero-divisor graph for finite rings, preprint 2005.
[12]S. P. Redmond. Central sets and radii of the zero-divisor graphs of commutative rings[J]. Comm. Algebra,2006,34,2389-2401.
[13]S. Akbari, H. R. Maimani, S. Yassemi. When a zero-divisor graph is planar or a completer r-partite graph[J]. J. Algebar,2003,270(1):169-180.
[14]F. DeMeyer, K. Schneider, Automorphisms and zero-divisor graphs of commutative rings[J], Internat. J. Commutative Rings,2002,1(3):93-106.
[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]T. S. Wu. On directed zero-divisor graphs of finite rings[J]. Discrete Math,2005,296(1): 73-86.
[17]S. Akbari, A. Mohammadian. Zero-divisor graphs of non-commutative rings[J], J. Algebra, 2006,296(2):462-479.
[18]G. A. Cannon, K. M. Neuerburg, S. P. Redmond. Zero-divisor graph of near rings and semigourp, Nearring and Nearfielas[J]. Dordrecht:Springer,189-200.
[19]S. P. Redmond. Structure in the zero-divisor graph of a noncommutative ring[J]. Houston J. Math,2004,30(2),345-355.
[20]C. P. Milies, S. K. Sehgal, An introduction to Group Rings[M], Kluwer Acadimic Publishers, Dordrecht,2002:125-150.
[21]J.A.邦迪,U.S.R.然蒂(著).吴望名,李念组,吴兰芳,谢伟如,梁文沛(译).图论及其应用[M].北京：科学出版社,1984：161-165.
[22]潘承洞,潘承彪(著).初等数论(第二版)[M].北京：北京大学出版社,1992：129-130.
[23]杨子胥(著).近似代数(第二版)[M].北京：高等教育出版社,2003.
[24]Brace A.Magurn. An Algebraic Introduction to K-Theory[M]. Cambridge University Press, 2002:120-125.
[25]R. K. Sharma, J. B.S rivastava and Manju Kbian. The unit group of FA4[J]. Publ:Math Debrecen.2007,71:21-27.
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