群环Z_nG的理想化零因子图的性质
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摘要
研究环的零因子图,以图的方式清晰、直观地刻画环的零因子的结构,这对理解环的结构本身具有重要意义。本文主要讨论了群环Z_nG关于增广理想Δ(G)的理想化Z_nG(+)Δ(G)的零因子图的性质,分别给出了环Z_nG(+)Δ(G)的零因子图的围长、直径和平面性的详细刻画,其中G为素数阶群。
It is very important to understand the structure of the ring itself by studying the zero-divisor graph of a ring to clearly and intuitively describe the structure of its zero-divisors by means of graph.Let Gbe a cyclic group of prime order,Z_n G group rings of Gover Z_n and Δ(G)augmentation ideals of Z_nG.Properties of zero-divisor graphs of idealizations of Z_n G with respect toΔ(G)are discussed in this paper.It provides detailed descriptions of the girth,the diameter and the planarity of zero-divisor graphs of idealizations of Z_n G,respectively.
It is very important to understand the structure of the ring itself by studying the zero-divisor graph of a ring to clearly and intuitively describe the structure of its zero-divisors by means of graph.Let Gbe a cyclic group of prime order,Z_n G group rings of Gover Z_n and Δ(G)augmentation ideals of Z_nG.Properties of zero-divisor graphs of idealizations of Z_n G with respect toΔ(G)are discussed in this paper.It provides detailed descriptions of the girth,the diameter and the planarity of zero-divisor graphs of idealizations of Z_n G,respectively.
引文
[1]ANDERSON D F,LIVINGSTON P S.The zero-divisor graph of a commutative ring[J].J Algebra,1999,217(2):434-447.DOI:10.1006/jabr.1998.7840.
[2]AXTEL M,STICKLES J.Zero-divisor graphs of idealizations[J].J Pure Appl Algebra,2006,204(2):235-243.DOI:10.1016/j.jpaa.2005.04.004.
[3]MAIMANI H R,YASSEMI S.Zero-divisor graphs of amalgamated duplication of a ring along an ideal[J].J Pure Appl Algebra,2008,212(1):168-174.DOI:10.1016/j.jpaa.2007.05.015.
[4]LUCAS T G.The diameter of a zero divisor graph[J].J Algebra,2006,301(1):174-193.DOI:10.1016/j.jalgebra.2006.01.019.
[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.DOI:10.1016/j.jpaa.2006.10.007.
[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(1):169-180.DOI:10.1016/S0021-8693(03)00370-3.
[7]BELSHOFF R,CHAPMAN J.Planar zero-divisor graphs[J].J Algebra,2007,316(1):471-480.DOI:10.1016/j.jalgebra.2007.01.049.
[8]AKBARIA S,MOHAMMADIAN A.On the zero-divisor graph of a commutative ring[J].J Algebra,2004,274(2):847-855.DOI:10.1016/S0021-8693(03)00435-6.
[9]ANDERSON D F,BADAWI A.The total graph of a commutative ring[J].J Algebra,2008,320(7):2706-2719.DOI:10.1016/j.jalgebra.2008.06.028.
[10]郭述锋,谢光明,易忠.群环ZnG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.DOI:10.16088/j.issn.1001-6600.2015.02.011.
[11]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.DOI:10.16088/j.issn.1001-6600.2009.02.004.
[12]MILIES C P,SEHGAL S K.An Introduction to Group Rings[M].Dordrecht:Kluwer Academic Publishers,2002.
[13]BUCKLEY F,LEWINTER M.图论简明教程[M].李慧霸,王风芹,译.北京:清华大学出版社,2005.
[2]AXTEL M,STICKLES J.Zero-divisor graphs of idealizations[J].J Pure Appl Algebra,2006,204(2):235-243.DOI:10.1016/j.jpaa.2005.04.004.
[3]MAIMANI H R,YASSEMI S.Zero-divisor graphs of amalgamated duplication of a ring along an ideal[J].J Pure Appl Algebra,2008,212(1):168-174.DOI:10.1016/j.jpaa.2007.05.015.
[4]LUCAS T G.The diameter of a zero divisor graph[J].J Algebra,2006,301(1):174-193.DOI:10.1016/j.jalgebra.2006.01.019.
[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.DOI:10.1016/j.jpaa.2006.10.007.
[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(1):169-180.DOI:10.1016/S0021-8693(03)00370-3.
[7]BELSHOFF R,CHAPMAN J.Planar zero-divisor graphs[J].J Algebra,2007,316(1):471-480.DOI:10.1016/j.jalgebra.2007.01.049.
[8]AKBARIA S,MOHAMMADIAN A.On the zero-divisor graph of a commutative ring[J].J Algebra,2004,274(2):847-855.DOI:10.1016/S0021-8693(03)00435-6.
[9]ANDERSON D F,BADAWI A.The total graph of a commutative ring[J].J Algebra,2008,320(7):2706-2719.DOI:10.1016/j.jalgebra.2008.06.028.
[10]郭述锋,谢光明,易忠.群环ZnG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.DOI:10.16088/j.issn.1001-6600.2015.02.011.
[11]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.DOI:10.16088/j.issn.1001-6600.2009.02.004.
[12]MILIES C P,SEHGAL S K.An Introduction to Group Rings[M].Dordrecht:Kluwer Academic Publishers,2002.
[13]BUCKLEY F,LEWINTER M.图论简明教程[M].李慧霸,王风芹,译.北京:清华大学出版社,2005.