剩余类环上二阶对称矩阵模的保行列式的加法映射
|
摘要
为了研究剩余类环上对称矩阵模的保行列式的加法映射,首先说明这类加法映射其实都是线性的,然后通过合同变换,利用数论知识和行列式运算并借助于整数的标准素分解进行分类讨论,以确定主要基底的像,再利用映射的线性性质确定所有矩阵的像,并讨论了本质上属于同一类映射的映射形式之间的关系。结果表明,剩余类环上二阶对称矩阵模上保行列式的加法映射都是规范的。研究方法解决了一般环上非零元未必有逆的本质带来的困难,将基础集扩展到剩余类环上,此结果可以看作是保行列式问题向环靠近的一小步,改进了线性保持问题的已有结果,对剩余类环上的其他保持问题的研究也具有参考价值。
In order to characterize the additive maps preserving of modulus of symmetric matrices over residue class rings,these maps are firstly proved to be linear in fact,then they are classified and discussed by means of contract transformation,number theory knowledge,determinant operation,and standard prime factorization of integers,to determine the image of the main base,and thus characterize the image of all matrices using the linearity.The relationship between the maps which have different forms but belong to the same class in fact is also discussed.The results show that additive maps preserving determinant on modulus of symmetric matrices over residue class rings are all trival.The research method solves the difficulty caused by the fact that non-zero elements in a general ring are not necessarily invertible,and extends the basic set to the residue class rings.This result can be regarded as a small step toward determinant preserving problem in a ring,which improves the existing results of the linear preserving problem.It has reference value for the study of other preserving problems on the remaining class rings.
In order to characterize the additive maps preserving of modulus of symmetric matrices over residue class rings,these maps are firstly proved to be linear in fact,then they are classified and discussed by means of contract transformation,number theory knowledge,determinant operation,and standard prime factorization of integers,to determine the image of the main base,and thus characterize the image of all matrices using the linearity.The relationship between the maps which have different forms but belong to the same class in fact is also discussed.The results show that additive maps preserving determinant on modulus of symmetric matrices over residue class rings are all trival.The research method solves the difficulty caused by the fact that non-zero elements in a general ring are not necessarily invertible,and extends the basic set to the residue class rings.This result can be regarded as a small step toward determinant preserving problem in a ring,which improves the existing results of the linear preserving problem.It has reference value for the study of other preserving problems on the remaining class rings.
引文
[1] FROBENIUS G.ber die Darstellung Der Endlichen Gruppen Durch Lineare Substitutionen[M].Berlin:Sitzungsber Deutsch Akad Wiss,1897.
[2] EATON M L.On linear transformations which preserve the determinant[J].Illinois Journal of Mathematics,1969,13(4):722-727.
[3]LAUTEMANN C.Linear transformations on matrices:Rank preservers and determinant preservers[J].Linear and Multilinear Algebra,1981,10(4):343-345.
[4] DOLINAR G,EMRL P.Determinant preserving maps on matrix algebras[J].Linear Algebra and Its Applications,2002,348(1/2/3):189-192.
[5] TAN V,WANG F.On determinant preserver problems[J].Linear Algebra and Its Applications,2003,369(1):311-317.
[6] CAO Chongguang,TANG Xiaomin.Determinant preserving transformations on symmetric matrix spaces[J].Electronic Journal of Linear Algebra,2004,11(1):205-211.
[7] ZHANG Xian,TANG Xiaomin,CAO Chongguang.Preserver Problems on Spaces of Matrices[M].Beijing:Science Press,2007.
[8] HUANG Huajun,LIU C N,SZOKOL P,et al.Trace and determinant preserving maps of matrices[J].Linear Algebra and Its Applications,2016,507(15):373-388.
[9] GERGO N.Determinant preserving maps:An infinite dimensional version of a theorem of frobenius[J].Linear and Multilinear Algebra,2017,65(2):351-360.
[10]MARCELL G,SOUMYASHANT N.On a class of determinant preserving maps for finite von Neumann algebras[J].Jouranl of Mathematical Analysis and Applications,2018,464(1):317-327.
[11]GOLBERG M A.The derivative of a determinant[J].American Mathematical Monthly,1972,79(10):1124-1126.
[12]PIERCE S.A survey of linear preserver problems[J].Pacific Journal of Mathematics,1992,204(2):257-271.
[13]AUPETIT B.Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras[J].Journal of the London Mathematical Society,2000,62(3):917-924.
[14]GUTERMAN A,LI C K,EMRL P.Some general techniques on linear preserver problems[J].Linear Algebra and Its Applications,2000,315(1/3):61-81.
[15]LI C K,TSING N K.Linear preserver problems:A brief introduction and some special techniques[J].Linear Algebra and Its Applications,1992,162(2):217-235.
[16]LI C K,PIERSE S.Linear preserver problems[J].American Mathematical Monthly,2001,108(7):591-605.
[17]EMRL P.Maps on matrix spaces[J].Linear Algebra and Its Applications,2006,413(2):364-393.
[18]DUFFNER M A,CRUZ H F D.Rank nonincreasing linear maps preserving the determinant of tensor product of matrices[J].Linear Algebra and Its Applications,2016,510(1):186-191.
[19]DING Yuting,FOSNER A,XU Jinli,et al.Linear maps preserving determinant of tensor products of Hermitian matrices[J].Journal of Mathematical Analysis and Applications,2016,446(2):1139-1153.
[20]HARDY Y,FOSNER A.Linear maps preserving kronecker quotients[J].Linear Algebra and Its Applications,2018,556(2):200-209.
[21]JI Youqing,LIU Ting,ZHU Sen.On linear maps preserving complex symmetry[J].Journal of Mathematical Analysis and Applications,2018,468(1):1144-1163.
[22]COSTARA C.Linear surjective maps preserving at least one element from the local spectrum[J].Proceedings of the Edinburgh Mathematical Society,2018,61(1):169-175.
[23]ZHANG Jiayu,SHENG Yuqiu.Additive maps preserving determinant on modules of matrices over[J].International Research Journal of Pure Algebra,2017,7(4):513-521.
[2] EATON M L.On linear transformations which preserve the determinant[J].Illinois Journal of Mathematics,1969,13(4):722-727.
[3]LAUTEMANN C.Linear transformations on matrices:Rank preservers and determinant preservers[J].Linear and Multilinear Algebra,1981,10(4):343-345.
[4] DOLINAR G,EMRL P.Determinant preserving maps on matrix algebras[J].Linear Algebra and Its Applications,2002,348(1/2/3):189-192.
[5] TAN V,WANG F.On determinant preserver problems[J].Linear Algebra and Its Applications,2003,369(1):311-317.
[6] CAO Chongguang,TANG Xiaomin.Determinant preserving transformations on symmetric matrix spaces[J].Electronic Journal of Linear Algebra,2004,11(1):205-211.
[7] ZHANG Xian,TANG Xiaomin,CAO Chongguang.Preserver Problems on Spaces of Matrices[M].Beijing:Science Press,2007.
[8] HUANG Huajun,LIU C N,SZOKOL P,et al.Trace and determinant preserving maps of matrices[J].Linear Algebra and Its Applications,2016,507(15):373-388.
[9] GERGO N.Determinant preserving maps:An infinite dimensional version of a theorem of frobenius[J].Linear and Multilinear Algebra,2017,65(2):351-360.
[10]MARCELL G,SOUMYASHANT N.On a class of determinant preserving maps for finite von Neumann algebras[J].Jouranl of Mathematical Analysis and Applications,2018,464(1):317-327.
[11]GOLBERG M A.The derivative of a determinant[J].American Mathematical Monthly,1972,79(10):1124-1126.
[12]PIERCE S.A survey of linear preserver problems[J].Pacific Journal of Mathematics,1992,204(2):257-271.
[13]AUPETIT B.Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras[J].Journal of the London Mathematical Society,2000,62(3):917-924.
[14]GUTERMAN A,LI C K,EMRL P.Some general techniques on linear preserver problems[J].Linear Algebra and Its Applications,2000,315(1/3):61-81.
[15]LI C K,TSING N K.Linear preserver problems:A brief introduction and some special techniques[J].Linear Algebra and Its Applications,1992,162(2):217-235.
[16]LI C K,PIERSE S.Linear preserver problems[J].American Mathematical Monthly,2001,108(7):591-605.
[17]EMRL P.Maps on matrix spaces[J].Linear Algebra and Its Applications,2006,413(2):364-393.
[18]DUFFNER M A,CRUZ H F D.Rank nonincreasing linear maps preserving the determinant of tensor product of matrices[J].Linear Algebra and Its Applications,2016,510(1):186-191.
[19]DING Yuting,FOSNER A,XU Jinli,et al.Linear maps preserving determinant of tensor products of Hermitian matrices[J].Journal of Mathematical Analysis and Applications,2016,446(2):1139-1153.
[20]HARDY Y,FOSNER A.Linear maps preserving kronecker quotients[J].Linear Algebra and Its Applications,2018,556(2):200-209.
[21]JI Youqing,LIU Ting,ZHU Sen.On linear maps preserving complex symmetry[J].Journal of Mathematical Analysis and Applications,2018,468(1):1144-1163.
[22]COSTARA C.Linear surjective maps preserving at least one element from the local spectrum[J].Proceedings of the Edinburgh Mathematical Society,2018,61(1):169-175.
[23]ZHANG Jiayu,SHENG Yuqiu.Additive maps preserving determinant on modules of matrices over[J].International Research Journal of Pure Algebra,2017,7(4):513-521.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |