模的w-相对性研究
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摘要
本文主要利用相对w-包络和w-模理论的方法研究任意的R-模,得到了一些较w-模更为一般的性质和结果.论文分为两章.在第一章中,我们首先引入了相对w-子模和相对w-包络的概念.并对其性质做了深入的研究,给出了相对w-子模成为准素子模的充要条件是(A:M)w≠R或者对于任意的J∈GV(R).有J (?)(A：M).通过局部化的方法我们得到：有限型模可以表示为它的某个有限生成子模的相对w-包络,并且有极大相对w-子模.接着证明了本章的主要定理——有限型模成为w-Xoether模当且仅当它有相对w-子模的升链条件.第二章的内容主要是围绕w-多余子模展开的.首先我们证得：单模与GV-挠模是GV-不可约模：w-Artin模的每个相对w-子模都包含一个GV-不可约子模.且w-Artin模关于极大w-理想的局部化恰好为Artin模.进而分别讨论了模的w-多余子模和w-Jacobson根.并给出了相应的例子.最后我们证得：对于满足条件每一个非平凡的相对w-子模包含在一个极大相对w-子模中的模.其w-多余子模与w-Jacobson根的子模是一一对应的.
In this paper, the methods of w-envelopes and w-module theory are employed to investigate an arbitrary R-module.We obtain some more general properties and results than w-module.This thesis is divided into two chapters.In Chapter 1.we firstly introduce two notions:relative w-submodule and relative w-envelope. and discuss some properties of them. Then we get that sufficient and necessary conditions for a primary submodule to be a relative w-submodule is (A:M)w,≠R or for an arbitrary GV-ideal.J∈(A:M).By the way of localization we obtain that any finite type module can be expressed as the relative w-envelope of one of its finite submodules and always has maximal relative w-submodule.At last,the main theorem of this chapter is shown that a finite type module is a w-Noether module if and only if it has the ascending chain condition on relative w-submodule.In Chapter 2.we do our work around w-superfluous submodules.We prove that simple module and GV-torsion module are GV-irreducible modules,each real relative w-submodule of w-Artin module includes a GV-irreducible submodule, and its localization about w-ideal is Artin-module. Moreover,we discussed w-superfluous submodules and w-Jacobson radicals separately. At the same time corresponding examples are also given.Finally, it is found that for the module satisfying the condition that each real relative w-submodule contained in a maximal relative w-submodule.there is a one to one correspondence between w-superfluous submodules and the submodules of w-Jacobson radical.
In this paper, the methods of w-envelopes and w-module theory are employed to investigate an arbitrary R-module.We obtain some more general properties and results than w-module.This thesis is divided into two chapters.In Chapter 1.we firstly introduce two notions:relative w-submodule and relative w-envelope. and discuss some properties of them. Then we get that sufficient and necessary conditions for a primary submodule to be a relative w-submodule is (A:M)w,≠R or for an arbitrary GV-ideal.J∈(A:M).By the way of localization we obtain that any finite type module can be expressed as the relative w-envelope of one of its finite submodules and always has maximal relative w-submodule.At last,the main theorem of this chapter is shown that a finite type module is a w-Noether module if and only if it has the ascending chain condition on relative w-submodule.In Chapter 2.we do our work around w-superfluous submodules.We prove that simple module and GV-torsion module are GV-irreducible modules,each real relative w-submodule of w-Artin module includes a GV-irreducible submodule, and its localization about w-ideal is Artin-module. Moreover,we discussed w-superfluous submodules and w-Jacobson radicals separately. At the same time corresponding examples are also given.Finally, it is found that for the module satisfying the condition that each real relative w-submodule contained in a maximal relative w-submodule.there is a one to one correspondence between w-superfluous submodules and the submodules of w-Jacobson radical.
引文
[1]Wang Fanggui,R.L.McCasland.On w-modules over Strong Mori domains. Comm.Algebra,1997.25(4):1285-1306.
[2]R. Gilmer.Multiplicative Ideal Theory. New York:Marcel Dekker INC,1972.
[3]L.Fuchs. L.Salce.Modules over Valuation Dom,ains. New York:Marcel Dekker INC.1985.
[4]J.L. Mott, M.Zafrullah.On Prufer u-multiplication domains. Manuscripta Math.,1981.35:1-26.
[5]J.R.Hedstrom.E.G.Houston.Some remarks on star-operations. J.Pure Appl. Algebra,1980.18:37-44.
[6]工芳贵.交换环与星型算子理论.北京：科学出版社.2006.
[7]王芳贵.星型算子理论的发展及其应用.四川师范大学学报：自然科学版.2009.32(2)：249-259.
[8]YIN H Y.WAGN F G,ZHU X S,CHEN Y H.w-Modules over commutative rings.to appear.
[9]WANG F G,ZHANG J.Injective modules over w-noetherian rings.to appear.
[10]WANG F G.On w-coherent domains.to appear.
[11]WANG F G.w-Projective over commutative rings.to appear.
[12]王芳贵,张俊.w-半单性.待发.
[13]张俊.硕士学位论文.
[14]D.D.Anderson, M.Zafrullah.On t-invertibility Ⅲ. Comm.Algebra,1993, 21(4):1189-1201.
[15]M.Zafrullah.Flatness and invertibility of an ideal. Comm.Algebra,1990. 18(7):2151-2158.
[16]Wang Fanggui.w-Dimension of domains. Comm.Algebra,1999,27(5):2267-2276.
[17]Wang Fanggui. w-Dimension of domains II. Comm. Algebra.2001.29(6): 2419-2428.
[18]Wang Fanggui. On w-projective modules and w-flat modules. Algebra Colloq.. 1997.4:111-120.
[19]Orzech M.Onto endmorphisms are isomorphisms.Amer. Math. Monthly.1971. 78:357-362.
[20]Frank W.Anderson.Kent R.Fuller.Rings and Categories of Modules. Second Edition.Berlin and New York,Springer-Verlag.1992.
[21]冯克勤.交换代数基础.北京：科学出版社.1986.
[22]陈晋健,陈顺卿.模论.开封：河南大学出版社.1994.
[23]D.E.Dobbs,E.G.Houston, T.G.Lucas. M.Roitman and M.Zafrullah.On t-linked overrings. Comm.Algebra,1992.20(5):1463-1488.
[24]H.Matsumura. Commutative Ring Theory.Cambridge:Cambridge University Press.1980.
[25]S.Glaz.Commutative Coherent Rings. Lecture Notes in Mathematics,no. 1371.Berlin and New York, Springer-Verlag,1989.
[26]Wang Fanggui.R.L.McCasland.On strong Mori domains. J.Pure Appl.Al-gebra.1999,135:155-165.
[27]J.D.Sally.W.V.Vasconcelos.Flat ideals I. Comm.Algebra,1975.3(6):531-543.
[28]Wang Fanggui.w-modules over a PVMD. Proceedings of the International Symposium on Teaching and Applications of Engineering Mathematics.Hong Kong,2001,117-120.
[29]J.J.Rotman.An Introduction to Homological Algebra. New York:Academic Press,1979.
[30]Wang Fanggui.On induced operations and UMT-domains. J.Sichuan Normal Univ.(Natural Science),2004,27(1):1-9.
[31]M.H.Park. Group rings and semigroup rings over strong Mori domains. J. Pure Appl.Algebra.2001,163:301-318.
[32]E.Houston,M. Zafrullah. Integral domains in which each t-ideal is divisorial. Michigan Math.J.,1988,35:291-300.
[33]J.L.Mott,M.Zafrullah.On Krull domains. Arch.Math.,1991,56:559-568.
[34]E.Kunz.Introduction to Commutative Algebra and Algebraic Geometry. Boston:Birkhauser,1985.
[35]J.W. Brewer.W.J.Heinzer.Associated primes of principal ideals. Duke Math. J.,1974.41:1-7.
[36]B.G.Kang. Prufer v-multiplication domains and the ring R[X]Nv.J.Algebra. 1989.123:151-170.
[37]工芳贵.GCD整环与UFD分式理想的刻划.科学通报.1997.42(5)：556.
[38]Wang Fanggui.On t-dimension over strong Mori domains. Acta Mathematica Sinica. English Series.2006.22(1):131-138.
[39]N.Jacobson.Basic Algebra Ⅰ. San Francisco:W.H.Freeman,1974.
[40]N.Jacobson.Basic Algebra Ⅱ. San Francisco:W.H.Freeman,1980.
[41]E.G.Houston.T.G.Lucas.T.M.Viswanathan.Primary decomposition of di-visorial ideals in Mori domains. J.Algebra.1988.117:327-342.
[42]M.F.Jones.M.L.Teply. Coherent rings of finite weak global dimension. Comm. Algebra,1982.10(5):493-503.
[43]V.Barucci,D.F.Anderson,D.E.Dobbs.Coherent Mori domains and the prin-cipal ideal theorem.Comm.Algebra,1987,15(6):1119-1156.
[44]M.Auslander.D.A.Buchsbaum.Homological dimension in Noetherian rings Ⅱ. Trans.Amer.Math. Soc.,1958,88:194-206.
[2]R. Gilmer.Multiplicative Ideal Theory. New York:Marcel Dekker INC,1972.
[3]L.Fuchs. L.Salce.Modules over Valuation Dom,ains. New York:Marcel Dekker INC.1985.
[4]J.L. Mott, M.Zafrullah.On Prufer u-multiplication domains. Manuscripta Math.,1981.35:1-26.
[5]J.R.Hedstrom.E.G.Houston.Some remarks on star-operations. J.Pure Appl. Algebra,1980.18:37-44.
[6]工芳贵.交换环与星型算子理论.北京：科学出版社.2006.
[7]王芳贵.星型算子理论的发展及其应用.四川师范大学学报：自然科学版.2009.32(2)：249-259.
[8]YIN H Y.WAGN F G,ZHU X S,CHEN Y H.w-Modules over commutative rings.to appear.
[9]WANG F G,ZHANG J.Injective modules over w-noetherian rings.to appear.
[10]WANG F G.On w-coherent domains.to appear.
[11]WANG F G.w-Projective over commutative rings.to appear.
[12]王芳贵,张俊.w-半单性.待发.
[13]张俊.硕士学位论文.
[14]D.D.Anderson, M.Zafrullah.On t-invertibility Ⅲ. Comm.Algebra,1993, 21(4):1189-1201.
[15]M.Zafrullah.Flatness and invertibility of an ideal. Comm.Algebra,1990. 18(7):2151-2158.
[16]Wang Fanggui.w-Dimension of domains. Comm.Algebra,1999,27(5):2267-2276.
[17]Wang Fanggui. w-Dimension of domains II. Comm. Algebra.2001.29(6): 2419-2428.
[18]Wang Fanggui. On w-projective modules and w-flat modules. Algebra Colloq.. 1997.4:111-120.
[19]Orzech M.Onto endmorphisms are isomorphisms.Amer. Math. Monthly.1971. 78:357-362.
[20]Frank W.Anderson.Kent R.Fuller.Rings and Categories of Modules. Second Edition.Berlin and New York,Springer-Verlag.1992.
[21]冯克勤.交换代数基础.北京：科学出版社.1986.
[22]陈晋健,陈顺卿.模论.开封：河南大学出版社.1994.
[23]D.E.Dobbs,E.G.Houston, T.G.Lucas. M.Roitman and M.Zafrullah.On t-linked overrings. Comm.Algebra,1992.20(5):1463-1488.
[24]H.Matsumura. Commutative Ring Theory.Cambridge:Cambridge University Press.1980.
[25]S.Glaz.Commutative Coherent Rings. Lecture Notes in Mathematics,no. 1371.Berlin and New York, Springer-Verlag,1989.
[26]Wang Fanggui.R.L.McCasland.On strong Mori domains. J.Pure Appl.Al-gebra.1999,135:155-165.
[27]J.D.Sally.W.V.Vasconcelos.Flat ideals I. Comm.Algebra,1975.3(6):531-543.
[28]Wang Fanggui.w-modules over a PVMD. Proceedings of the International Symposium on Teaching and Applications of Engineering Mathematics.Hong Kong,2001,117-120.
[29]J.J.Rotman.An Introduction to Homological Algebra. New York:Academic Press,1979.
[30]Wang Fanggui.On induced operations and UMT-domains. J.Sichuan Normal Univ.(Natural Science),2004,27(1):1-9.
[31]M.H.Park. Group rings and semigroup rings over strong Mori domains. J. Pure Appl.Algebra.2001,163:301-318.
[32]E.Houston,M. Zafrullah. Integral domains in which each t-ideal is divisorial. Michigan Math.J.,1988,35:291-300.
[33]J.L.Mott,M.Zafrullah.On Krull domains. Arch.Math.,1991,56:559-568.
[34]E.Kunz.Introduction to Commutative Algebra and Algebraic Geometry. Boston:Birkhauser,1985.
[35]J.W. Brewer.W.J.Heinzer.Associated primes of principal ideals. Duke Math. J.,1974.41:1-7.
[36]B.G.Kang. Prufer v-multiplication domains and the ring R[X]Nv.J.Algebra. 1989.123:151-170.
[37]工芳贵.GCD整环与UFD分式理想的刻划.科学通报.1997.42(5)：556.
[38]Wang Fanggui.On t-dimension over strong Mori domains. Acta Mathematica Sinica. English Series.2006.22(1):131-138.
[39]N.Jacobson.Basic Algebra Ⅰ. San Francisco:W.H.Freeman,1974.
[40]N.Jacobson.Basic Algebra Ⅱ. San Francisco:W.H.Freeman,1980.
[41]E.G.Houston.T.G.Lucas.T.M.Viswanathan.Primary decomposition of di-visorial ideals in Mori domains. J.Algebra.1988.117:327-342.
[42]M.F.Jones.M.L.Teply. Coherent rings of finite weak global dimension. Comm. Algebra,1982.10(5):493-503.
[43]V.Barucci,D.F.Anderson,D.E.Dobbs.Coherent Mori domains and the prin-cipal ideal theorem.Comm.Algebra,1987,15(6):1119-1156.
[44]M.Auslander.D.A.Buchsbaum.Homological dimension in Noetherian rings Ⅱ. Trans.Amer.Math. Soc.,1958,88:194-206.