群环Z_nG的基于理想△(G)的零因子图
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摘要
主要讨论了群环Z_nG的基于理想△(G)的零因子图Γ_(△(G))(Z_nG)的性质,分别给出了Γ_(△(G))(Z_nG)的围长,平面性和直径的详细刻画.同时,给出了交换环R基于其理想I的零因子图Γ_I(R)与商环R/I的零因子图Γ(R/I)的直径的关系的一个刻画.
Let G be a finite abelian group of order at least 2,Z_nG be group rings of G over Z_n and △(G) be augmentation ideals of Z_nG.Properties of zero-divisor graphs of Z_nG based on △(G) are mainly discussed in this paper.Characterizations on the girth,the planarity and the diameter of zero-divisor graphs of Z_nG based on A(G) are given,respectively.Moreover,the relationship between diameters of zero-divisor graphs of R based on I and diameters of zero-divisor graphs of residue class rings R/I is given,where R is a commutative ring with identity and I is a nonzero proper ideal of R.
Let G be a finite abelian group of order at least 2,Z_nG be group rings of G over Z_n and △(G) be augmentation ideals of Z_nG.Properties of zero-divisor graphs of Z_nG based on △(G) are mainly discussed in this paper.Characterizations on the girth,the planarity and the diameter of zero-divisor graphs of Z_nG based on A(G) are given,respectively.Moreover,the relationship between diameters of zero-divisor graphs of R based on I and diameters of zero-divisor graphs of residue class rings R/I is given,where R is a commutative ring with identity and I is a nonzero proper ideal of R.
引文
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[8]郭述锋,谢光明,易忠.群环Z_nG的零因子图的性质[J].广西师范大学学报:自然科学版,2015,33(2):68-75.
[9]郭述锋,徐承杰,易忠.群环Z_nG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.
[10]黄基荣,郭述锋.群环RG_p的代数性质及其结构[J].数学的实践与认识,2015,45(3):268-272.
[11]Milies C P,Sehgal S K.An Introduction to Group Rings[M].Dordrecht:Kluwer Academic Publishers,2002.
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[2]Redmond S P.An ideal-based zero-divisor graph of a commutative ring[J].Comm.Algebra,2003,31(9):4425-4443.
[3]Osba E A,Al-Addasi S,Jaradeh N A.Zero divisor graph for the ring of gaussian integers modulo n[J].Comm.Algebra,2008,36(10):3865-3877.
[4]Lucas T G.The diameter of a zero divisor graph[J].J.Algebra,2006,301:174-193.
[5]Anderson D F,Mulay S B.On the diameter and girth of a zero-divisor graph[J],J.Pure Appl.Algebra,2007,210(2):543-550.
[6]Akbari S,Maimani H R,Yassemi S.When a zero-divisor graph is planar or a complete r-partite graph[J].J.Algebra,2003,270:169-180.
[7]Belshoff R,Chapman J.Planar zero-divisor graphs[J].J.Algebra,2007,316:471-480.
[8]郭述锋,谢光明,易忠.群环Z_nG的零因子图的性质[J].广西师范大学学报:自然科学版,2015,33(2):68-75.
[9]郭述锋,徐承杰,易忠.群环Z_nG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.
[10]黄基荣,郭述锋.群环RG_p的代数性质及其结构[J].数学的实践与认识,2015,45(3):268-272.
[11]Milies C P,Sehgal S K.An Introduction to Group Rings[M].Dordrecht:Kluwer Academic Publishers,2002.
[12]Buckley F,Lewinter M(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.