2-强Gorenstein半单环上模的结构及其应用
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摘要
研究了局部2-强Gorenstein半单环上任一模M的结构,证明了M可以唯一分解为不可分解模的直和。利用模M的直和分解,引入了有限生成模M的秩rank(M)的概念,证明了在有限局部2-强Gorenstein半单环上这样定义的秩就是线性码的信息位数。
The structure of the module Mover the local 2-strongly Gorenstein semisimple ring is investigated. Namely,Mis uniquely decomposed into a direct sum of indecomposable modules. By the decomposition of Minto direct sum,the definition of the rank of finitely generated module Mis introduced. It is proved that,the rank defined over the local 2-strongly Gorenstein semisimple ring is the information bit of the linear codes.
The structure of the module Mover the local 2-strongly Gorenstein semisimple ring is investigated. Namely,Mis uniquely decomposed into a direct sum of indecomposable modules. By the decomposition of Minto direct sum,the definition of the rank of finitely generated module Mis introduced. It is proved that,the rank defined over the local 2-strongly Gorenstein semisimple ring is the information bit of the linear codes.
引文
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[2]ENOCHS E E,HUANG Zhaoyong.Canonical filtrations of Gorenstein injective modules[J].Proc Amer Math Soc,2011,139(7):2415-2421.
[3]YOSHINO Y.Cohen-Macaulay modules over Cohen-Macaulay rings[M].Cambridge:Cambridge University Press,1990.
[4]BENNIS D,HU Kui,WANG Fanggui.On 2-SG-semisimple rings[J].Rocky Mountain J Math,2015,45:1093-1100.
[5]BENNIS D,MAHDOU N,OUARGHI K.Rings over which all modules are strongly Gorenstein projective[J].Rocky Mountain J M ath,2007,40(3):749-759.
[6]AUSLANDER M,BRIDGER M.Stable module theory[M].Providence,RI:Amer Math Soc,1969.
[7]BENNIS D.(n,m)-SG rings[J].Mathematics,2009,35(2D):169-178.
[8]BENNIS D,MAHDOU N.A generalization of strongly Gorenstein projective modules[J].J Algebra Appl,2009,8(2):219-227.
[9]BENNIS D,MAHDOU N.Strongly Gorenstein projective,injective and flat modules[J].J Algebra Appl,2007,210(4):437-445.
[10]ENOCHS E E,JENDA O M G.Gorenstein injective and projective modules[J].Math Z,1995,220(1):611-633.
[11]KASCH F,WALLACE D A R.Modules and rings[M].New York:Academic Press,1982.
[12]ROTMAN,JOSEPH J.An introduction to homological algebra[M].New York:Academic Press,INC.1979.
[13]ABUALRUB T,GHRAYEB A,OEHMKE R H.The rank of Z4cyclic codes of length 2e[C]//First International Symposium on Control,Communications and Signal Processing.[S.l.]:IEEE,2004:651-654.
[14]DOUGHERTY S T,SHIROMOTO K.Maximum distance codes over rings of order 4[J].IEEE Tran Inf Theory,2001,47(1):400-404.
[15]ZHU Shixin,SHI Minjia.The ranks of cyclic and negacyclic codes over the finite ring R[J].Chin J Electron,2008,25(1):97-101.
[16]BONNECAZE A,UDAYA P.Cyclic codes and self-dual codes over F2+u F2[J].IEEE Tran Inf Theory,1999,45(4):1250-1255.
[17]ZHU Shixin,LI Yan,DENG Lin.A class of constacyclic MDS codes over Fq+u Fq+…+us-1Fq[J].Journal of University of Science and Technology of China,2013,3(3):197-201.