关于Г-环上元素的Drazin逆
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摘要
本文分四个部分:
第一章回顾了广义逆发展概况以及至今一些重要研究成果,同时介绍了Γ-环研究发展概况.
第二章介绍了本文所用到的一些基本概念,相关的基本理论知识.
第三章给出Γ-环上元素的Γ_(αβ)-Drazin逆,讨论Γ_(αβ)-Drazin逆存在唯一性和Γ_(αβ)-Drazin逆与加权Drazin逆之间的一些关系,进一步探讨Γ_α-Drazin逆的一些性质和Γ-环上元素的α-核心-幂零分解定理,最后讨论Γ-环上元素Γ_α-群逆,给出Γ_α-群逆存在的充要条件及一些表达式.
第四章研究了任意Γ-环上元素的Γ_α~(Dk)逆存在性问题,而且,对于存在p'和q',使得p'αpαa=a=aαqαq'式成立时,乘积pαaαq的Γ_α~(Dk)逆可以得出表达式且容易计算出来.那么,对于这种乘积的群逆结果就很容易推出.这些结果也可以推广到范畴的态射中.
In this dissertation,
Chapter 1 is devoted to review the history and development of the generalize inverse and gamma ring.
Chapter 2 introduces some conceptions related to soliton theory.
Chapter 3 introduces theΓ_(α，β)- Drazin inverses of the elements over gamma ring . Existence and uniqueness of theΓ_(α,β)- Drazin inverses of the elements and the relationship between theΓ_(α,β)- Drazin inverses and the weighted Drazin inverses are discussed. Some properties of theΓ_α- Drazin inverse and theα-core-nilpotent decoposition theorem for the elements over gamma ring are given. TheΓ_α- group inverses of the elements over gamma ring are discussed. Some necessary and sufficient conditions for existance and some expression ofΓ_α- group inverses are given.
Chapter 4 gives the characterizations for existance of theΓ_α~(Dk)-inverse of an elements over an arbitray gamma ring. Moreover,theΓ_α~(Dk)-inverse of a product pαaαq for which there exist a p' and q' such that p'αpαa =a= aαqαq' can be characterized and computed. This generalizes recent results obtained for the group inverse of such products. This results also apply to morphisms in(additive)categories.
第一章回顾了广义逆发展概况以及至今一些重要研究成果,同时介绍了Γ-环研究发展概况.
第二章介绍了本文所用到的一些基本概念,相关的基本理论知识.
第三章给出Γ-环上元素的Γ_(αβ)-Drazin逆,讨论Γ_(αβ)-Drazin逆存在唯一性和Γ_(αβ)-Drazin逆与加权Drazin逆之间的一些关系,进一步探讨Γ_α-Drazin逆的一些性质和Γ-环上元素的α-核心-幂零分解定理,最后讨论Γ-环上元素Γ_α-群逆,给出Γ_α-群逆存在的充要条件及一些表达式.
第四章研究了任意Γ-环上元素的Γ_α~(Dk)逆存在性问题,而且,对于存在p'和q',使得p'αpαa=a=aαqαq'式成立时,乘积pαaαq的Γ_α~(Dk)逆可以得出表达式且容易计算出来.那么,对于这种乘积的群逆结果就很容易推出.这些结果也可以推广到范畴的态射中.
In this dissertation,
Chapter 1 is devoted to review the history and development of the generalize inverse and gamma ring.
Chapter 2 introduces some conceptions related to soliton theory.
Chapter 3 introduces theΓ_(α，β)- Drazin inverses of the elements over gamma ring . Existence and uniqueness of theΓ_(α,β)- Drazin inverses of the elements and the relationship between theΓ_(α,β)- Drazin inverses and the weighted Drazin inverses are discussed. Some properties of theΓ_α- Drazin inverse and theα-core-nilpotent decoposition theorem for the elements over gamma ring are given. TheΓ_α- group inverses of the elements over gamma ring are discussed. Some necessary and sufficient conditions for existance and some expression ofΓ_α- group inverses are given.
Chapter 4 gives the characterizations for existance of theΓ_α~(Dk)-inverse of an elements over an arbitray gamma ring. Moreover,theΓ_α~(Dk)-inverse of a product pαaαq for which there exist a p' and q' such that p'αpαa =a= aαqαq' can be characterized and computed. This generalizes recent results obtained for the group inverse of such products. This results also apply to morphisms in(additive)categories.
引文
[1] Fredholm I.Sur une classe d'equations fonctionnelles[M],Acta Math.,1903(27),pp.365-390.
[2] Hilbert D.Grundzug einer allgeneinen Theorie der linearen Integraleichun-gen[M],B.G.Teubner,Leipzig and Berlin,1912,Reprint of six articles which appeared originally in the Gottingen Nachrichten,1904,pp. 49-51;1904,pp.213-259;1905,pp.307-338;1906,pp.l57-227;1906,pp.439-480;1910,pp.355-417.
[3] Reid W.Generalized Green's Matrices for compatible systems of differential equa-tions[M],Amer.J.Math., 1931(53),pp.443-459.
[4] Ben-Israel A. Greville T.N.E. Generalized Inverses;Theory and applications[M],John Wiley, New York, 1974.
[5] Moore E.H. On the reciprocal of the general algebraic matrix[M],Bull.Amer.Math.Soc,1920(26),pp.394-395.
[6] Groetsch C.W. Generalized Inverses of linear Operators:Representation and Approximation[M],Dekker, New York,1977.
[7] Penrose R. A generalized inverse for matrices[J],Proc.Cambridge Philos.Soc.,1955(51),pp.406-413.
[8] Campbell S.L.,Meyer C.D. Generalized Inverses of Linear Transformations[M],Pitman,Lndon,1979.
[9] Clien R.E., Greville T.N.E.A Drazin inverse for retangular matrices[J],L.A.A.,1980(29),pp.53-62.
[10] Rao C.R.,Mitra S.K. Generalized Inverse of Matrices and Its Applications[M],John Wiley,New York, 1971.
[11] Campbell S.L.(Ed.).Recent Applications of Generalized Inverses[M],Pitman,London,1982.
[12] Drazin M.P. Pseudoinverses in associative rings and semigroups[J],Amer.Math.monthly,1958(65):506-514.
[13] Greville T.N.E.Spectral generalized inverses of squares matrices[J],MRC Technical Science Rep. 823,Mathematics Research Center,Univ.of Wisconsin,Madison,1967.
[14] 王国荣.矩阵与算子广义逆[M].北京:科学出版社,1994
[15] 陈建龙.关于环上矩阵的群逆与Drazin逆[J].数学学报,1994(37):373-380
[16] Nashed M.Z.Generalized inverses and applications[M], Academic Press, New York,1976.
[17] Wilkinson J.H. Note on the pratical significance of the Drazin inverse,in Recent Applications of Generalized Inverses[M],S.L.Campbell,Ed.,Pitman.London, 1982.
[18] Bhaskara Rao K.P.S. The theory of generalized inverses over commutative rings[M],Algebra,logic and Applications series Volume 17,Tylor and Francis,London and Neew York,2002.
[19] Clien R E,Greville T N E. A Drazin inverse for ractangular matrices[J].Linear Algebra Appl,1980(21):53-62.
[20] 岑建苗.关于长方矩阵的F-Drazin逆和柱心-幂零分解[J].高等学校计算数学学报,2006,28(3):216-223.
[21] 岑建苗.关于长方矩阵的加权群逆得存在性[J].计算数学,2007,89(1):39-48.
[22] 岑建苗.环上矩阵广义Moore-Penrose逆得存在性[J].数学学报,2006,49(3):544-558.
[23] 岑建苗.矩阵Γ_α-Moore Penrose逆与星序[J].工程数学学报,2006(23):127-132.
[24] 岑建苗.范畴中态射集的Γ-减序[J].数学研究与评论,2005(25):683-690.
[25] Hartwig R E,Puystjens R.The group inverse of a companion matrics[J] .Linear and Multi-Linear Algebra,1997(43):137-150
[26] Patricio P.Puystjens R,Drazin-Moore-Penrose invertibility in rings [J] .Linear Algebra Appl,2004(389):159-173.
[27] Puystjens R. ,Gouveia M.C.Drazin invertibility for matrices over an arbitrary ring[J].Linear Algebra Appl, 2004(385):105-116.
[28] Rakocevic V,Wei Y. A weighted drazin inverse and application [J].Linear Algebra Appl,2002(350):25-39.
[29] Kyuno S. Gamma rings[M] . Mathematics .U.S.A:Hadrcnic press,Inc, 1985.
[30] Barnes W E. On the Γ-rings of Nobusawa[M].Pacific J Math,1966(18):411-422.
[31] Coppage W E.and Luh J. Radicals of gamma rings[M].Math Soc Japan,1971(23):40-52.
[32] Kyuno S. A gamma ring with the right and left unities[M]. Math. Jacobson ,1979(24):191-193.
[33] Chen.W.X. The largest Von Neumann regular ideal of a gamma ring[J].Zhejiang Univ,1984(18):133-138.
[34] Chen.W.X. On strong nilpotency of Γ-rings and generalized Γ-rings [J].Zhejiang Univ,1985(19):114-116.
[35] Nobusawa N. On a generralization of the ring theory[M].Osaka . Math.l964(l),81-89.
[36] Booth G.L. Radicals of matrix gamma rings[M].Math.Japonica,1988,33(3):325-334.
[37] Ravisanker T.S.and Shukla U.S. Structure of Γ-rings[M].Pacific Jour.Math,1979,80(2):537-559.
[38] Cen Jianmiao.Matrix Γ_(n,m)-ring over Γ-ring[J].Ningbo Normal College,1989(6):1-5.
[39] 奚欧根.Γ-环与广义Γ-环的强幂零根与拟幂零根[J].数学研究与评论,1988,8(2):171-174.
[40] 陈维新.Γ-环的单位元[J].数学研究与评论,1991,11(3):319-324.
[41] 张寿传,陈维新.Γ-环的一般根论和Baer根[J].浙江大学学报,1991,25(6):719-724.
[42] 陈维新.Γ-环及其矩阵环的QN-根和K-根[J].数学研究与评论,1997,17(2):214-216.
[43] 陈维新.Γ-环元素的强幂零性所确定的根[J].数学年刊,1993,14A(2):255-261.
[44] 陈维新,张寿传.Γ-环的Jacobson根和幂等心根[J].浙江大学学报,1993,27(4):431-437.
[45] 刘邵学.环与代数[M].科学出版社,1983.
[2] Hilbert D.Grundzug einer allgeneinen Theorie der linearen Integraleichun-gen[M],B.G.Teubner,Leipzig and Berlin,1912,Reprint of six articles which appeared originally in the Gottingen Nachrichten,1904,pp. 49-51;1904,pp.213-259;1905,pp.307-338;1906,pp.l57-227;1906,pp.439-480;1910,pp.355-417.
[3] Reid W.Generalized Green's Matrices for compatible systems of differential equa-tions[M],Amer.J.Math., 1931(53),pp.443-459.
[4] Ben-Israel A. Greville T.N.E. Generalized Inverses;Theory and applications[M],John Wiley, New York, 1974.
[5] Moore E.H. On the reciprocal of the general algebraic matrix[M],Bull.Amer.Math.Soc,1920(26),pp.394-395.
[6] Groetsch C.W. Generalized Inverses of linear Operators:Representation and Approximation[M],Dekker, New York,1977.
[7] Penrose R. A generalized inverse for matrices[J],Proc.Cambridge Philos.Soc.,1955(51),pp.406-413.
[8] Campbell S.L.,Meyer C.D. Generalized Inverses of Linear Transformations[M],Pitman,Lndon,1979.
[9] Clien R.E., Greville T.N.E.A Drazin inverse for retangular matrices[J],L.A.A.,1980(29),pp.53-62.
[10] Rao C.R.,Mitra S.K. Generalized Inverse of Matrices and Its Applications[M],John Wiley,New York, 1971.
[11] Campbell S.L.(Ed.).Recent Applications of Generalized Inverses[M],Pitman,London,1982.
[12] Drazin M.P. Pseudoinverses in associative rings and semigroups[J],Amer.Math.monthly,1958(65):506-514.
[13] Greville T.N.E.Spectral generalized inverses of squares matrices[J],MRC Technical Science Rep. 823,Mathematics Research Center,Univ.of Wisconsin,Madison,1967.
[14] 王国荣.矩阵与算子广义逆[M].北京:科学出版社,1994
[15] 陈建龙.关于环上矩阵的群逆与Drazin逆[J].数学学报,1994(37):373-380
[16] Nashed M.Z.Generalized inverses and applications[M], Academic Press, New York,1976.
[17] Wilkinson J.H. Note on the pratical significance of the Drazin inverse,in Recent Applications of Generalized Inverses[M],S.L.Campbell,Ed.,Pitman.London, 1982.
[18] Bhaskara Rao K.P.S. The theory of generalized inverses over commutative rings[M],Algebra,logic and Applications series Volume 17,Tylor and Francis,London and Neew York,2002.
[19] Clien R E,Greville T N E. A Drazin inverse for ractangular matrices[J].Linear Algebra Appl,1980(21):53-62.
[20] 岑建苗.关于长方矩阵的F-Drazin逆和柱心-幂零分解[J].高等学校计算数学学报,2006,28(3):216-223.
[21] 岑建苗.关于长方矩阵的加权群逆得存在性[J].计算数学,2007,89(1):39-48.
[22] 岑建苗.环上矩阵广义Moore-Penrose逆得存在性[J].数学学报,2006,49(3):544-558.
[23] 岑建苗.矩阵Γ_α-Moore Penrose逆与星序[J].工程数学学报,2006(23):127-132.
[24] 岑建苗.范畴中态射集的Γ-减序[J].数学研究与评论,2005(25):683-690.
[25] Hartwig R E,Puystjens R.The group inverse of a companion matrics[J] .Linear and Multi-Linear Algebra,1997(43):137-150
[26] Patricio P.Puystjens R,Drazin-Moore-Penrose invertibility in rings [J] .Linear Algebra Appl,2004(389):159-173.
[27] Puystjens R. ,Gouveia M.C.Drazin invertibility for matrices over an arbitrary ring[J].Linear Algebra Appl, 2004(385):105-116.
[28] Rakocevic V,Wei Y. A weighted drazin inverse and application [J].Linear Algebra Appl,2002(350):25-39.
[29] Kyuno S. Gamma rings[M] . Mathematics .U.S.A:Hadrcnic press,Inc, 1985.
[30] Barnes W E. On the Γ-rings of Nobusawa[M].Pacific J Math,1966(18):411-422.
[31] Coppage W E.and Luh J. Radicals of gamma rings[M].Math Soc Japan,1971(23):40-52.
[32] Kyuno S. A gamma ring with the right and left unities[M]. Math. Jacobson ,1979(24):191-193.
[33] Chen.W.X. The largest Von Neumann regular ideal of a gamma ring[J].Zhejiang Univ,1984(18):133-138.
[34] Chen.W.X. On strong nilpotency of Γ-rings and generalized Γ-rings [J].Zhejiang Univ,1985(19):114-116.
[35] Nobusawa N. On a generralization of the ring theory[M].Osaka . Math.l964(l),81-89.
[36] Booth G.L. Radicals of matrix gamma rings[M].Math.Japonica,1988,33(3):325-334.
[37] Ravisanker T.S.and Shukla U.S. Structure of Γ-rings[M].Pacific Jour.Math,1979,80(2):537-559.
[38] Cen Jianmiao.Matrix Γ_(n,m)-ring over Γ-ring[J].Ningbo Normal College,1989(6):1-5.
[39] 奚欧根.Γ-环与广义Γ-环的强幂零根与拟幂零根[J].数学研究与评论,1988,8(2):171-174.
[40] 陈维新.Γ-环的单位元[J].数学研究与评论,1991,11(3):319-324.
[41] 张寿传,陈维新.Γ-环的一般根论和Baer根[J].浙江大学学报,1991,25(6):719-724.
[42] 陈维新.Γ-环及其矩阵环的QN-根和K-根[J].数学研究与评论,1997,17(2):214-216.
[43] 陈维新.Γ-环元素的强幂零性所确定的根[J].数学年刊,1993,14A(2):255-261.
[44] 陈维新,张寿传.Γ-环的Jacobson根和幂等心根[J].浙江大学学报,1993,27(4):431-437.
[45] 刘邵学.环与代数[M].科学出版社,1983.