模的内射性及相互关系
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摘要
本论文主要讨论了内射模的各种推广以及它们之间的一些关系。本论文共分三章。
第一章是引言,介绍了模的内射性几种推广形式的研究背景、研究历史、现状以及研究意义,其中主要包括GQP-内射模、AQP-内射模、AGP-内射模、APQ-内射模、QPQ-内射模。
第二章中探讨了几种模的内射性的推广,介绍了拟GP-内射模、主拟内射模(PQ-内射模)、拟AP-内射模、几乎主拟内射模(APQ-内射模)、几乎拟主内射模(AQP-内射模)、拟主拟内射模(QPQ-内射模)、GQP-内射模的定义及其相应的等价刻画；与此同时,我们还探讨了内射模推广形式之间的相互关系,从QP-内射模及GQP-内射模的定义我们知道QP-内射模一定是GQP-内射模并且环R是右P-内射环当且仅当模RR是QP-内射模,R是GP-内射环当且仅当模RR是GQP-内射模。由于GP-内射环不一定是P-内射环,所以GQP-内射模未必是QP-内射模。
第三章中探讨了两类特殊内射模GQP-内射模及AQP-内射模的一些性质,一个模的GQP-内射性以及它的自同态环的PQ-内射性、AQP-内射性以及它的AP-内射性之间的关系,从研究W(S)与J(S)的关系介绍了GQP-内射模与AQP-内射模所共有的性质,另一方面研究了S与W(S)中元素的关系以及非零M-循环子模在其中所起的串联作用。
This thesis mainly investigated the properties of some classes of special injective modules and their relationships.
The first chapter introduced the concepts of injective modules, such as GQP-injective modules, AQP-injective modules, AGP-injective modules, APQ-injective modules、QPQ-injective modules. And we recall some valuable results on them.
In chapterⅡ, we introduced the definitions of quasi GP-injective modules、PQ-injective modules、quasi AP-injective modules、APQ-injective modules、AQP-injective modules、QPQ-injective modules、GQP-injective modules, and discussed their relations. By definitions of QP-injective modules and GQP-injective modules we know that the QP-injective modules are GQP-injective modules, the ring R is right P-injective ring if and only if the module RR is QP-injective module, and the ring R is GP-injective ring if and only if the module RR is GQP-injective module. Since GP-injective rings are not necessarily P-injective ring, GQP-injective module may not be QP-injective module.
The two special injective modules, i.e. GQP-injective modules and AQP-injective modules, are dicussed in the third chapter. Meanwhile, we investigated the GQP-injectivity of a module and the PQ-injectivity of it's endomorphism ring, the AQP-injectivity of a module and the AP-injectivity of it's endomorphism ring. Through the study of W(S) and J(S),we know that the GQP-injective modules and AQP-injective modules shared the same property. Furthermore, we realized the M-cyclic submodule paly an important role by investigating the elements of S and W(S).
第一章是引言,介绍了模的内射性几种推广形式的研究背景、研究历史、现状以及研究意义,其中主要包括GQP-内射模、AQP-内射模、AGP-内射模、APQ-内射模、QPQ-内射模。
第二章中探讨了几种模的内射性的推广,介绍了拟GP-内射模、主拟内射模(PQ-内射模)、拟AP-内射模、几乎主拟内射模(APQ-内射模)、几乎拟主内射模(AQP-内射模)、拟主拟内射模(QPQ-内射模)、GQP-内射模的定义及其相应的等价刻画；与此同时,我们还探讨了内射模推广形式之间的相互关系,从QP-内射模及GQP-内射模的定义我们知道QP-内射模一定是GQP-内射模并且环R是右P-内射环当且仅当模RR是QP-内射模,R是GP-内射环当且仅当模RR是GQP-内射模。由于GP-内射环不一定是P-内射环,所以GQP-内射模未必是QP-内射模。
第三章中探讨了两类特殊内射模GQP-内射模及AQP-内射模的一些性质,一个模的GQP-内射性以及它的自同态环的PQ-内射性、AQP-内射性以及它的AP-内射性之间的关系,从研究W(S)与J(S)的关系介绍了GQP-内射模与AQP-内射模所共有的性质,另一方面研究了S与W(S)中元素的关系以及非零M-循环子模在其中所起的串联作用。
This thesis mainly investigated the properties of some classes of special injective modules and their relationships.
The first chapter introduced the concepts of injective modules, such as GQP-injective modules, AQP-injective modules, AGP-injective modules, APQ-injective modules、QPQ-injective modules. And we recall some valuable results on them.
In chapterⅡ, we introduced the definitions of quasi GP-injective modules、PQ-injective modules、quasi AP-injective modules、APQ-injective modules、AQP-injective modules、QPQ-injective modules、GQP-injective modules, and discussed their relations. By definitions of QP-injective modules and GQP-injective modules we know that the QP-injective modules are GQP-injective modules, the ring R is right P-injective ring if and only if the module RR is QP-injective module, and the ring R is GP-injective ring if and only if the module RR is GQP-injective module. Since GP-injective rings are not necessarily P-injective ring, GQP-injective module may not be QP-injective module.
The two special injective modules, i.e. GQP-injective modules and AQP-injective modules, are dicussed in the third chapter. Meanwhile, we investigated the GQP-injectivity of a module and the PQ-injectivity of it's endomorphism ring, the AQP-injectivity of a module and the AP-injectivity of it's endomorphism ring. Through the study of W(S) and J(S),we know that the GQP-injective modules and AQP-injective modules shared the same property. Furthermore, we realized the M-cyclic submodule paly an important role by investigating the elements of S and W(S).
引文
[1]Chen jianlong and Ding nanqing. On general principally injective rings [J]. Comm. Algebra,1999,27(5):2097-2116.
[2]Nicholson W K,Park,J K,Yousif,M F.Principally injective rings[J]Journal of algebra,1995,174(2):77-93.
[3]Sanh N V,Shum K P,Dhompongsa S,Wongwai S.On Quasiprincipally injective modules[J].Algebra Colloquium,1999,6(3)269-276
[4]Zhao Yu-e,Du xian neng.On quasi GP-injective modules[J]. College Mathematics 2005,21(3):38-41.
[5]Nam S B,Kim N K,Kim J Y.On simple GP-injective modules[J]. Comm.Algebra, 1995,23(14):5437-5444.
[6]Zhang jule,Wu jun.Generalizations of principally injective rings[J]. Algebra Colloqium,1999,6(3):277-282.
[7]Zhao Yu-e,Du xian neng.On quasi GP-injective modules [J] Journal of Anhui university Natural Science Edition 2007,31(4).
[8]W.K.Nicholson and M.F.Yousif,Principally injective rings, J.Algebra 174(1995),No.1,77-93.
[9]W.K.Nicholson,J.K.Park,and M.F.Yousif,Principally quasi-injective modules Comm. Algebra27(1999),no.4,1683-1693.
[10]N.V.Sanh,K.P.Shum,S.Dhompongsa,and S.Wongwai,On quasi-Principally injective modules,Algebra Colloq.6(1999),no.3,269-276.
[11]Zhao Yu-e,Du Xian-neng, Some studies on Quasi AP-injective Modules.Northeast.Math.J.22(1)(2006),73-80.
[12]S.S.Page and Y.Zhou,Generalizations of principally injective rings,J.Algebra206(1998),No.2,706-721.
[13]Z.M.Zhu,Z.S.Xia and Z.S.Tan,Generalizations of principa-Lly quasi-injective modules and quasiprincipally injective modules,Int.J.Math.Sci.(12)(2005),1853-1860.
[14]Some studies on quasi AP-injective modules.Northeast.math J.22(1)(2006),73-80.
[15]Z.M.Zhu,On general quasi-principally injective modules, Southeast Asian Bull. Math.30(2)(2006),391-397.
[16]Anderson.F.W.and Fuller.K.R.,Rings and Categories of Modules[M],New York, Springer2Verlag,1974
[17]Zhu Z M. Some results on GQP-injective modules [J].International electronic journal of algebra,2009,5:1-6.
[18]Du xian-neng,Zhao chun-e,Pseudo-Injective modules and Principally pseudo-injective modules. Journal of mathematical research and exposition27(2),2007.
[19]Wisbauer,R.,Foundations of Module and Ring Theory,Gordon and Breach,London-Tokyo,1991.
[20]Sanh N V and Shum K P.Endomorphism Rings of quasiprincipally injective modules[J].Comm.in Algebra,2001,29(4) 1437-1443
[21]Nicholson W K,Yousif M F.On perfect simple-injective rings[J].Proc.Amer.Math.Soc.,1997,125(4):979-985.
[22]Ding N Q,Yousif M F,and Zhou Y Q,Modules with annihilator conditions[J].Comm.Algebra,2002,30(5):2309-2320.
[2]Nicholson W K,Park,J K,Yousif,M F.Principally injective rings[J]Journal of algebra,1995,174(2):77-93.
[3]Sanh N V,Shum K P,Dhompongsa S,Wongwai S.On Quasiprincipally injective modules[J].Algebra Colloquium,1999,6(3)269-276
[4]Zhao Yu-e,Du xian neng.On quasi GP-injective modules[J]. College Mathematics 2005,21(3):38-41.
[5]Nam S B,Kim N K,Kim J Y.On simple GP-injective modules[J]. Comm.Algebra, 1995,23(14):5437-5444.
[6]Zhang jule,Wu jun.Generalizations of principally injective rings[J]. Algebra Colloqium,1999,6(3):277-282.
[7]Zhao Yu-e,Du xian neng.On quasi GP-injective modules [J] Journal of Anhui university Natural Science Edition 2007,31(4).
[8]W.K.Nicholson and M.F.Yousif,Principally injective rings, J.Algebra 174(1995),No.1,77-93.
[9]W.K.Nicholson,J.K.Park,and M.F.Yousif,Principally quasi-injective modules Comm. Algebra27(1999),no.4,1683-1693.
[10]N.V.Sanh,K.P.Shum,S.Dhompongsa,and S.Wongwai,On quasi-Principally injective modules,Algebra Colloq.6(1999),no.3,269-276.
[11]Zhao Yu-e,Du Xian-neng, Some studies on Quasi AP-injective Modules.Northeast.Math.J.22(1)(2006),73-80.
[12]S.S.Page and Y.Zhou,Generalizations of principally injective rings,J.Algebra206(1998),No.2,706-721.
[13]Z.M.Zhu,Z.S.Xia and Z.S.Tan,Generalizations of principa-Lly quasi-injective modules and quasiprincipally injective modules,Int.J.Math.Sci.(12)(2005),1853-1860.
[14]Some studies on quasi AP-injective modules.Northeast.math J.22(1)(2006),73-80.
[15]Z.M.Zhu,On general quasi-principally injective modules, Southeast Asian Bull. Math.30(2)(2006),391-397.
[16]Anderson.F.W.and Fuller.K.R.,Rings and Categories of Modules[M],New York, Springer2Verlag,1974
[17]Zhu Z M. Some results on GQP-injective modules [J].International electronic journal of algebra,2009,5:1-6.
[18]Du xian-neng,Zhao chun-e,Pseudo-Injective modules and Principally pseudo-injective modules. Journal of mathematical research and exposition27(2),2007.
[19]Wisbauer,R.,Foundations of Module and Ring Theory,Gordon and Breach,London-Tokyo,1991.
[20]Sanh N V and Shum K P.Endomorphism Rings of quasiprincipally injective modules[J].Comm.in Algebra,2001,29(4) 1437-1443
[21]Nicholson W K,Yousif M F.On perfect simple-injective rings[J].Proc.Amer.Math.Soc.,1997,125(4):979-985.
[22]Ding N Q,Yousif M F,and Zhou Y Q,Modules with annihilator conditions[J].Comm.Algebra,2002,30(5):2309-2320.