R（+）M与R（？）I的零因子图
摘要
有限交换环一直是代数学中的不可或缺的重要研究对象,它不仅内涵丰富,而且在众多数学分支(如:组合数学,有限几何和算法分析)以及工程科学(如:编码理论和序列设计)中有着重要的应用.特别是最近二三十年中,由于有限环在编码领域中的作用越来越突出,使得对有限环的研究成果更加丰富.用图的性质研究代数结构是最近20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.它主要涉及数学中的六个领域:环论,群论,半群论,图论,初等数论和组合数学,如此众多的交叉研究,使它不但具有趣味性和吸引力,而且有广阔的发展前景.近年,它已成为国际上的一个热门研究领域.
在本文的第一章中我们概述了零因子图发展的历史,本文的研究背景以及本文的主要结果.同时我们还给出了环论与图论的一些基本的概念和结论.
第二章中我们研究了任意交换环的idealization的零因子图的平面性,得到了一些较满意的结果.
在本文的第三章中,我们讨论了idealization的零因子图的围长.对有限交换环的idealization的零因子图的围长给出了比[28]更具体的刻划.
在第四章中我们研究了amalgamated duplication of a commutative ring along an ideal的零因子图的平面性,得到了一些较满意的结果,且讨论了它的零因子图的直径.
在本文的第五章中我们讨论了交换环的idealization的基于理想的零因子图.特别地,我们刻划了它的零因子图的围长与直径.
The theory of finite commutative rings is a very active area which is not only of great theoreticalinterest in itself but also found important applications both within mathematics (for instance,in Combinatorics, Finite Geometries and the Analysis Algorithms) and within the EngineeringSciences (in particular in Coding Theory and Sequence Design). especially,in recent decades,there are more and more abundant research results of finite ring because of the finite ring play amore and more prominent role in coding areas. The study of algebraic structures, using properties ofgraphs, has become an exciting research topic in the last twenty years, leading to many fascinatingresults and questions. It lies at the crossroads of six areas: ring theory, group theory, semigrouptheory, graph theory, elementary number theory and combinatorics. Like so many interdisciplinarystudies, it has its fascinations, attractions, and also vast potential for future development. It hasbecome an active research topic in recent years.
In Chapter 1 of this paper, we summarize the history of the zero-divisor graphs, and give thebackground and main results of this paper. At the same time, we give the notation and basicresults of ring theory and graph theory.
In Chapter 2, we will determine the planarity of the zero-divisor graph of idealization of anycommutative ring and get some approving results. The main results of this chapter will be veryuseful in the following chapters.
In Chapter 3, we will study the girth of the zero-divisor graph of idealization, give more specificcharacterizations about the girth of the zero-divisor graph of idealization of any finite commutativerings than those given in [28].
In Chapter 4, we will determine the planarity of the zero-divisor graph of amalgamated du-plication of a commutative ring along an ideal and get some satisfying results, and also study thediameter of the zero-divisor graph.
In Chapter 5, we will discuss ideal-based zero-divisor graphs of idealization of any commutativerings. Specifically, we characterize the girth and diameter of the graphs.
在本文的第一章中我们概述了零因子图发展的历史,本文的研究背景以及本文的主要结果.同时我们还给出了环论与图论的一些基本的概念和结论.
第二章中我们研究了任意交换环的idealization的零因子图的平面性,得到了一些较满意的结果.
在本文的第三章中,我们讨论了idealization的零因子图的围长.对有限交换环的idealization的零因子图的围长给出了比[28]更具体的刻划.
在第四章中我们研究了amalgamated duplication of a commutative ring along an ideal的零因子图的平面性,得到了一些较满意的结果,且讨论了它的零因子图的直径.
在本文的第五章中我们讨论了交换环的idealization的基于理想的零因子图.特别地,我们刻划了它的零因子图的围长与直径.
The theory of finite commutative rings is a very active area which is not only of great theoreticalinterest in itself but also found important applications both within mathematics (for instance,in Combinatorics, Finite Geometries and the Analysis Algorithms) and within the EngineeringSciences (in particular in Coding Theory and Sequence Design). especially,in recent decades,there are more and more abundant research results of finite ring because of the finite ring play amore and more prominent role in coding areas. The study of algebraic structures, using properties ofgraphs, has become an exciting research topic in the last twenty years, leading to many fascinatingresults and questions. It lies at the crossroads of six areas: ring theory, group theory, semigrouptheory, graph theory, elementary number theory and combinatorics. Like so many interdisciplinarystudies, it has its fascinations, attractions, and also vast potential for future development. It hasbecome an active research topic in recent years.
In Chapter 1 of this paper, we summarize the history of the zero-divisor graphs, and give thebackground and main results of this paper. At the same time, we give the notation and basicresults of ring theory and graph theory.
In Chapter 2, we will determine the planarity of the zero-divisor graph of idealization of anycommutative ring and get some approving results. The main results of this chapter will be veryuseful in the following chapters.
In Chapter 3, we will study the girth of the zero-divisor graph of idealization, give more specificcharacterizations about the girth of the zero-divisor graph of idealization of any finite commutativerings than those given in [28].
In Chapter 4, we will determine the planarity of the zero-divisor graph of amalgamated du-plication of a commutative ring along an ideal and get some satisfying results, and also study thediameter of the zero-divisor graph.
In Chapter 5, we will discuss ideal-based zero-divisor graphs of idealization of any commutativerings. Specifically, we characterize the girth and diameter of the graphs.
引文
[1] I. Beck, Coloring of commutative ring [J], J. Algebra 116(1988)208-226.
[2] D. D. Anderson, M. Naseer, Beck s coloring of a commutative ring [J], J. algebra 159(1993)500-514.
[3] D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring [J], J. Algebra 217(1999)434-447.
[4] M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutativerings [J], Comm. Algebra 33(6)(2005), 2043-2050.
[5] S. B. Mulay, Cycles and symmetries of zero-divisors [J], Comm. Algebra 30(7)(2002), 3553-3558.
[6] R. Akhtar, L. Lee, Connectivity of the zero-divisor graph for fnite rings, preprint 2005.
[7] F. DeMeyer, K. Schneider, Automorphisms and zero divisor graphs of commu- tative rings, Commutativerings [J], Internet. J. Commutative Rings 2002, 1(3), 93-106.
[8] S. P. Redmond, Central sets and radii of the zero-divisor graphs of commutative rings [J], Comm. Algebra34(2006),2389-2401.
[9] A. Akbari, H. R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph [J],J. algebra 270(1)(2003) 169-180.
[10] H. Jen, C. Hsieh, H. J. Wang, Commutative ring with toroidal zero-divisor graphs [J], Arxiv, Preprint, mathAc 10702451 preprint 2007.
[11] H. J. Wang, Zero-divisor graphs of genus one [J], J. Algebra 304(2006), 666-678.
[12] D. F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras[J], J. Pure Appl. Algebra 180(2003) 221-241.
[13] R. Levy, J. Shapiro, The zero-divisor graph of von Neumann regular rings [J], Comm. Algebra 30(2)(2002),745-750.
[14] N. I. Cordova, C. Gholston, H. A. Hauser, The structure of zero-divisor graphs, preprint 2005.
[15] A. Phillips, J. Rogers, K. Tolliver, F. Worek, Uncharted territory of Zero divisor graphs and their complements[J], Miami university, SUMSRI 2004.
[16] K. Samei, The zero-divisor graph of a reduced ring [J], J. Pure Appl. Algebra 209(2007), 813-821.
[17] S. P. Redmond, On zero-divisor graph of small fnite commutative rings [J], Discrete Math 307(2007) 1155-1166.
[18] D. F. Anderson, S. B. Mulay. On the diameter and girth of a zero-divisor graph [J], J. Pure Appl. Algebra210 (2007) 543-550.
[19] J. D. LaGrange, Complemented Zero-Divisor Graphs and Boolean Rings [J], J.Algebra (2007), doi: 10. 1016/j.jalgebra. 2006.12.030.
[20] D. C. Lu, T. S. Wu, The zero-divisor graphs which are uniquely determined by neighborhoods [J], Comm.Algebra 35 (12) (2007), pp. 3855– 3864.
[21] H. J. Wang, Graphs associated to Co-maximal ideals of commutative rings [J], J. Algebra 320(2008) 2917-2933.
[22] T. S. Wu, D. C. Lu, The structure of a class of Z-local rings [J], Sci.China,Series A 49(10)(2006) 1297-1302.
[23] T. G. Lucas, The diameter of a zero divisor graph [J], J. Algebra 301(2006) 174-193.
[24] S. B. Mulay, Rings having Zero-divisor graphs of small diameter or large girth [J], Bull. Austral. Math. Soc72(2005), 481-490
[25] S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring [J], J. Algebra 274(2004)847-855.
[26] N. O. Smith, Planar zero-divisor graphs [J], Internet. J. Commutative Rings 2(4)2003, 177-188.
[27] M. Axtell, J. Stickles, J. Warfel, Zero-divisor graph of direct products of commutative ring [J], HoustonJournal of Mathematics (2006).
[28] M. Axtell, J. Stickles, Zero-divisor graphs of idealizations [J], J. Pure Appl. Algebra 204(2006) 235-243.
[29] D. C. Lu, W. T. Tong, The zero-divisor graphs of abelian regular rings [J], Northeast. Math. J 20(3)(2004),339-348.
[30] J. Warfel, Zero-divisor graphs for full matrix rings over finite fields [J], Linear. Algebra. Appl 428(2008)2947-2954.
[31] A. Abdollahi, Commutative graphs of commutative rings, Preprint2005.
[32] S. Akbari, M. Ghandehari, M. Hadian, A. Mohammadian, On commutative graphs of semisimple rings [J],Linear. Algebra. Appl 398(2004) 345-355.
[33] H. R. Maimani, S. Yassemi, Zero-divisor graphs of amalgamated duplication of a ring along an ideal [J], J.Pure Appl. Algebra 212(2008) 168-174.
[34] F. Azarpanah, M. Motamedi, Zero-divisor graph of C(x) [J], Acta Math. Hungra 108(1-2)(2005) 25-36.
[35] D. F. Anderson, A. Frazier, P. S. Livingston. The zero-divisor graph of a commutative ring, II [M], LectureNotes in Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York, 2001, PP. 61-72.
[36] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring [J], Comm. Algebra 31(9)(2003)4425-4442.
[37] D. C. Lu, T. S. Wu, On ideal-based zero-divisor graphs, Preprint 2006.
[38] H. R. Maimani, M. R. Pournaki, S. Yassemi, Zero-divisor graph with respect to an ideal [J], Comm. Algebra34(3)(2006), 923-929.
[39] S. P. Redmond, Regularity, completeness, and fniteness in the zero-divisor and ideal-based graphs of commu-tative rings [J], Internat. J. Commutative Rings (to appear).
[40] H. R. Maimani, M. R. Pournaki, S. Yassemi, zero-divisor graph with reapect to ideal [J], Comm. Algebra(2006).
[41] S. P. Redmond, The zero-divisor graph of a non-commutative ring [J], Internat. J. Commutative Rings1(4)(2002) 203-211.
[42] T. S. Wu, On directed zero-divisor graphs of fnite rings [J], Discrete Math 296(1)(2005) 73-86.
[43] S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings [J], J. Algebra 296(2)(2006)462-479.
[44] G. A. Cannon, K. M. Neuerburg, S. P. Redmond, Zero-divisor graph of nearrings and semigourp, Nearringand Nearfields [J], Dordrecht: Springer 189-200.
[45] S. P. Redmond, Structure in the zero-divisor graph of a noncommutative ring [J], Houston J. Math 30(2)(2004)345-355.
[46] F. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup [J], SemigroupForum 65(2)(2002) 206-214.
[47] F. DeMeyer, L. DeMeyer, Zero divisor graphs of semigroups [J], J. Algebra 283(1)(2005) 190-198.
[48] T. S. Wu, F. Cheng, The structures of zero-divisor semigroups with graph Kn ? K2 [J], Semigroup Forum76(2)(2008) 330-340.
[49] T. S. Wu, L. Chen, Simple graphs and commutative zero-divisor semigroups [J], Algebra. Colloq 16(2)(2009)211-218.
[50] T. S. Wu, D. C. Lu, Zero-divisor semigroups and some simple graphs [J], Comm. Algebra 34(8)(2006) 3043-3052.
[51] F. Demeyer, L. Demeyer, zero divisor graphs of Semigroups [J], J. Algebra 288(1)(2005) 190-198.
[52] Demeyer, Mckenie, Schneider, The Zero-divisor Graph of a Commutative semigroup [J], Semigroup Forum65(2)(2002) 206-214.
[53] T. S. Wu, D. C. Lu, Sub-semigroups determined by the zero-divisor graph [J], Discrete Math 308(22)(2008)5122-5135.
[54] Tang Gaohua, Su Huadong, Ren Beishang, Commutative Zero-Divisor Semigroups of Graphs With at mostFour Vertices [J], Algebra Colloquium 16(2)(2009) 341-350.
[55] T. S. Wu, Q. Liu, L. Chen, Zero divisor semigroups and renements of a star graph [J], Discrete Math309(8)(2009) 2510-2518 .
[56] P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings [J], J. Algebra 176(1995)124-127.
[57] H. R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings [J], J. Algebra319(2008) 1801-1808.
[58] S. Akbari, M. Ghandehari, M. Hadian, A. Mohammadian, On commuting graphs of semisimple rings [J],Linear Algebra and Its Applications 390(2004) 345-355.
[59] A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting graph of a group [J], J. Algebra 298(2006) 468-492.
[60] J. Coykendall, J. Maney, Irreducible divisor graphs [J], Comm. Algebra 35(3)(2007) 885-895.
[61] J. Maney, Irreducible divisor graphs II [J], Comm. Algebra 36(9)(2008) 3496-3513.
[62] P. W. Chen, A kind of graph structure of ring [J], Algebra Colloquium 10(2)(2003) 229-238.
[63] M. Zuo, The application of graph theory to a commutative semigroup [J], Journal of Shanghai institute ofTechnology 3(2003) 85-188.(In Chinese).
[64] M. Zuo, T. S. Wu, A new graph structure of commutative semigroups [J], Semigroup Forum 70(2005) 71-80.
[65] S. P. Redmond, Corrigendum to”On zero-divisor graphs of small fnite commutative rings”[Discrete Math.307(2007) 1155 - 1166] [J], Discrete Math 307(2007) 2449-2452.
[66] R. Belsho?, J. Chapman, Planar zero-divisor graphs [J], J. Algebra 316(2007) 471-480.
[67] M. Nagata, Local Rings [M], Interscience, New York, 1962.
[68] B. Corbas, G. D. Williams, Rings of order p5. I. Nonlocal rings [J], J. Algebra 231(2)(2000) 677-690.
[69] B. Corbas, G. D. Williams, Rings of order p5. II. Nonlocal rings [J], J. Algebra 231(2)(2000) 691-704.
[70] N. O. Smith, Infinite Planar Zero-divisor Graphs [J], Comm. Algebra 35(2007) 171-180.
[71] M. D. Anna, M. Fontana, An amalgamated duplication of a ring along an ideal [J], J. Algebra Appl(in press).
[72] F. DeMeyer, K. Schneider, Automorphisms and Zero-divisor Graphs of Commutative Rings [J], J. Algebra231(2) (2000) 691-704.
[2] D. D. Anderson, M. Naseer, Beck s coloring of a commutative ring [J], J. algebra 159(1993)500-514.
[3] D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring [J], J. Algebra 217(1999)434-447.
[4] M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutativerings [J], Comm. Algebra 33(6)(2005), 2043-2050.
[5] S. B. Mulay, Cycles and symmetries of zero-divisors [J], Comm. Algebra 30(7)(2002), 3553-3558.
[6] R. Akhtar, L. Lee, Connectivity of the zero-divisor graph for fnite rings, preprint 2005.
[7] F. DeMeyer, K. Schneider, Automorphisms and zero divisor graphs of commu- tative rings, Commutativerings [J], Internet. J. Commutative Rings 2002, 1(3), 93-106.
[8] S. P. Redmond, Central sets and radii of the zero-divisor graphs of commutative rings [J], Comm. Algebra34(2006),2389-2401.
[9] A. Akbari, H. R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph [J],J. algebra 270(1)(2003) 169-180.
[10] H. Jen, C. Hsieh, H. J. Wang, Commutative ring with toroidal zero-divisor graphs [J], Arxiv, Preprint, mathAc 10702451 preprint 2007.
[11] H. J. Wang, Zero-divisor graphs of genus one [J], J. Algebra 304(2006), 666-678.
[12] D. F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras[J], J. Pure Appl. Algebra 180(2003) 221-241.
[13] R. Levy, J. Shapiro, The zero-divisor graph of von Neumann regular rings [J], Comm. Algebra 30(2)(2002),745-750.
[14] N. I. Cordova, C. Gholston, H. A. Hauser, The structure of zero-divisor graphs, preprint 2005.
[15] A. Phillips, J. Rogers, K. Tolliver, F. Worek, Uncharted territory of Zero divisor graphs and their complements[J], Miami university, SUMSRI 2004.
[16] K. Samei, The zero-divisor graph of a reduced ring [J], J. Pure Appl. Algebra 209(2007), 813-821.
[17] S. P. Redmond, On zero-divisor graph of small fnite commutative rings [J], Discrete Math 307(2007) 1155-1166.
[18] D. F. Anderson, S. B. Mulay. On the diameter and girth of a zero-divisor graph [J], J. Pure Appl. Algebra210 (2007) 543-550.
[19] J. D. LaGrange, Complemented Zero-Divisor Graphs and Boolean Rings [J], J.Algebra (2007), doi: 10. 1016/j.jalgebra. 2006.12.030.
[20] D. C. Lu, T. S. Wu, The zero-divisor graphs which are uniquely determined by neighborhoods [J], Comm.Algebra 35 (12) (2007), pp. 3855– 3864.
[21] H. J. Wang, Graphs associated to Co-maximal ideals of commutative rings [J], J. Algebra 320(2008) 2917-2933.
[22] T. S. Wu, D. C. Lu, The structure of a class of Z-local rings [J], Sci.China,Series A 49(10)(2006) 1297-1302.
[23] T. G. Lucas, The diameter of a zero divisor graph [J], J. Algebra 301(2006) 174-193.
[24] S. B. Mulay, Rings having Zero-divisor graphs of small diameter or large girth [J], Bull. Austral. Math. Soc72(2005), 481-490
[25] S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring [J], J. Algebra 274(2004)847-855.
[26] N. O. Smith, Planar zero-divisor graphs [J], Internet. J. Commutative Rings 2(4)2003, 177-188.
[27] M. Axtell, J. Stickles, J. Warfel, Zero-divisor graph of direct products of commutative ring [J], HoustonJournal of Mathematics (2006).
[28] M. Axtell, J. Stickles, Zero-divisor graphs of idealizations [J], J. Pure Appl. Algebra 204(2006) 235-243.
[29] D. C. Lu, W. T. Tong, The zero-divisor graphs of abelian regular rings [J], Northeast. Math. J 20(3)(2004),339-348.
[30] J. Warfel, Zero-divisor graphs for full matrix rings over finite fields [J], Linear. Algebra. Appl 428(2008)2947-2954.
[31] A. Abdollahi, Commutative graphs of commutative rings, Preprint2005.
[32] S. Akbari, M. Ghandehari, M. Hadian, A. Mohammadian, On commutative graphs of semisimple rings [J],Linear. Algebra. Appl 398(2004) 345-355.
[33] H. R. Maimani, S. Yassemi, Zero-divisor graphs of amalgamated duplication of a ring along an ideal [J], J.Pure Appl. Algebra 212(2008) 168-174.
[34] F. Azarpanah, M. Motamedi, Zero-divisor graph of C(x) [J], Acta Math. Hungra 108(1-2)(2005) 25-36.
[35] D. F. Anderson, A. Frazier, P. S. Livingston. The zero-divisor graph of a commutative ring, II [M], LectureNotes in Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York, 2001, PP. 61-72.
[36] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring [J], Comm. Algebra 31(9)(2003)4425-4442.
[37] D. C. Lu, T. S. Wu, On ideal-based zero-divisor graphs, Preprint 2006.
[38] H. R. Maimani, M. R. Pournaki, S. Yassemi, Zero-divisor graph with respect to an ideal [J], Comm. Algebra34(3)(2006), 923-929.
[39] S. P. Redmond, Regularity, completeness, and fniteness in the zero-divisor and ideal-based graphs of commu-tative rings [J], Internat. J. Commutative Rings (to appear).
[40] H. R. Maimani, M. R. Pournaki, S. Yassemi, zero-divisor graph with reapect to ideal [J], Comm. Algebra(2006).
[41] S. P. Redmond, The zero-divisor graph of a non-commutative ring [J], Internat. J. Commutative Rings1(4)(2002) 203-211.
[42] T. S. Wu, On directed zero-divisor graphs of fnite rings [J], Discrete Math 296(1)(2005) 73-86.
[43] S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings [J], J. Algebra 296(2)(2006)462-479.
[44] G. A. Cannon, K. M. Neuerburg, S. P. Redmond, Zero-divisor graph of nearrings and semigourp, Nearringand Nearfields [J], Dordrecht: Springer 189-200.
[45] S. P. Redmond, Structure in the zero-divisor graph of a noncommutative ring [J], Houston J. Math 30(2)(2004)345-355.
[46] F. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup [J], SemigroupForum 65(2)(2002) 206-214.
[47] F. DeMeyer, L. DeMeyer, Zero divisor graphs of semigroups [J], J. Algebra 283(1)(2005) 190-198.
[48] T. S. Wu, F. Cheng, The structures of zero-divisor semigroups with graph Kn ? K2 [J], Semigroup Forum76(2)(2008) 330-340.
[49] T. S. Wu, L. Chen, Simple graphs and commutative zero-divisor semigroups [J], Algebra. Colloq 16(2)(2009)211-218.
[50] T. S. Wu, D. C. Lu, Zero-divisor semigroups and some simple graphs [J], Comm. Algebra 34(8)(2006) 3043-3052.
[51] F. Demeyer, L. Demeyer, zero divisor graphs of Semigroups [J], J. Algebra 288(1)(2005) 190-198.
[52] Demeyer, Mckenie, Schneider, The Zero-divisor Graph of a Commutative semigroup [J], Semigroup Forum65(2)(2002) 206-214.
[53] T. S. Wu, D. C. Lu, Sub-semigroups determined by the zero-divisor graph [J], Discrete Math 308(22)(2008)5122-5135.
[54] Tang Gaohua, Su Huadong, Ren Beishang, Commutative Zero-Divisor Semigroups of Graphs With at mostFour Vertices [J], Algebra Colloquium 16(2)(2009) 341-350.
[55] T. S. Wu, Q. Liu, L. Chen, Zero divisor semigroups and renements of a star graph [J], Discrete Math309(8)(2009) 2510-2518 .
[56] P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings [J], J. Algebra 176(1995)124-127.
[57] H. R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings [J], J. Algebra319(2008) 1801-1808.
[58] S. Akbari, M. Ghandehari, M. Hadian, A. Mohammadian, On commuting graphs of semisimple rings [J],Linear Algebra and Its Applications 390(2004) 345-355.
[59] A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting graph of a group [J], J. Algebra 298(2006) 468-492.
[60] J. Coykendall, J. Maney, Irreducible divisor graphs [J], Comm. Algebra 35(3)(2007) 885-895.
[61] J. Maney, Irreducible divisor graphs II [J], Comm. Algebra 36(9)(2008) 3496-3513.
[62] P. W. Chen, A kind of graph structure of ring [J], Algebra Colloquium 10(2)(2003) 229-238.
[63] M. Zuo, The application of graph theory to a commutative semigroup [J], Journal of Shanghai institute ofTechnology 3(2003) 85-188.(In Chinese).
[64] M. Zuo, T. S. Wu, A new graph structure of commutative semigroups [J], Semigroup Forum 70(2005) 71-80.
[65] S. P. Redmond, Corrigendum to”On zero-divisor graphs of small fnite commutative rings”[Discrete Math.307(2007) 1155 - 1166] [J], Discrete Math 307(2007) 2449-2452.
[66] R. Belsho?, J. Chapman, Planar zero-divisor graphs [J], J. Algebra 316(2007) 471-480.
[67] M. Nagata, Local Rings [M], Interscience, New York, 1962.
[68] B. Corbas, G. D. Williams, Rings of order p5. I. Nonlocal rings [J], J. Algebra 231(2)(2000) 677-690.
[69] B. Corbas, G. D. Williams, Rings of order p5. II. Nonlocal rings [J], J. Algebra 231(2)(2000) 691-704.
[70] N. O. Smith, Infinite Planar Zero-divisor Graphs [J], Comm. Algebra 35(2007) 171-180.
[71] M. D. Anna, M. Fontana, An amalgamated duplication of a ring along an ideal [J], J. Algebra Appl(in press).
[72] F. DeMeyer, K. Schneider, Automorphisms and Zero-divisor Graphs of Commutative Rings [J], J. Algebra231(2) (2000) 691-704.