有零因子的交换环上w-模的链条件及其应用
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摘要
本文研究了交换环上w-模的升链和降链条件,引入了几个新的模类,使得一些经典的理论得到新的表现和应用.我们证明了w-Noether环上有限型模的对偶模是有限表现型的.并讨论了w-Noether模的真w-子模的准素分解问题.也研究了w-Noether环上有限型w-模的零化子,证明了w-Noether环上有限型的GV -无挠模只有有限个极大素理想,且每一个都是其中某个非零元素的零化子.其次,我们引入了w-单模的概念,不仅说明了w-单模的存在性,而且指出了w-单模与单模的差异.进而,给出了w-半单模的定义.并得到了半单环的新的刻画.证明了R是半单环当且仅当每一个w-模是w-半单模.还讨论了w-半单模的w-子模链条件的等价刻画.同时,定义了w-模的w-底座.说明了w-底座与底座是不同的两个概念.并借助w-底座得到了w-Artin模的等价条件.此外,还给出了w-模的w-Jacobson根与w-多余子模两个新概念.我们也举例说明了w-Jacobson根与一般模范畴中定义的Jacobson根相比是非平凡的,并且讨论了两者之间的关系,证明了w(R) ? J(R).还得到了关于w-Jacobson根的中山引理的相应形式.作为w-Jacobson根的应用,证明了关于w-模的Kertese定理.另外,通过定义w-加性补,进一步研究了w-模的降链条件.最后,在Krull-Remak-Schmidt定理的观点下,讨论了几类模直和分解的唯一性问题.推广了Orzech定理,得到了更一般形式的Vasconcelos定理.还讨论了w-模的Fitting引理.证明了Schur引理对于w-单模仍然成立.进而,证明了w-半单模, w-Noether环上非零的GV -无挠的内射模以及有w-合成列的w-模可以唯一分解为自同态环是局部环的不可分解子模的直和.
In this paper, we investigate some properties of the ascending and descend-ing chain conditions of w-modules and introduce several new modules, so thatsome classical theories have new representation and application. Firstly, we provethat if R is a w-Noetherian ring, dual modules of finite type modules are finitelypresented type, and discuss the primary decomposition of w-submodules of w-Noetherian modules. We also study the annihilator of a non-zero element of finitetype w-modules. And it is proved that if R is a w-Noetherian ring and M is aGV -torsion-free R-module of finite type, then there exist only a finite number ofmaximal prime ideals, each of which is the annihilator of a non-zero element ofM. Secondly, we show the concept of w-simple modules. And we not only showthe existence of w-simple modules, but also point out the di?erence between w-simple modules and simple modules. Then the definition of the w-semisimplemodule is given. We also obtain new characterizations of semisimple rings andprove that R is semisimple if and only if every w-module is w-semisimple. Andwe discuss the equivalent characterizations of chain conditions of w-submodulesof w-semisimple modules. Meanwhile, we define the w-socle of a w-module, showthat the w-socle and the socle are di?erent and obtain equivalent conditions ofw-Artin modules by means of the w-socle. In addition, we show the concepts ofthe w-Jacobson radical of a w-module and the w-super?uous submodule. We alsogive an example to illustrate that in comparison to the Jacobson radical, the def-inition of the w-Jacobson radical is nontrivial and prove that w(R) ? J(R). Andwe obtain the Nakayama’s Lemma with respect to the w-Jacobson radical. Asthe application, the Kertese Theorem of w-modules is proved. By the definitionof w-addition complements, we character the descending chain conditions of w-modules. At last, we investigate the uniqueness problem of direct decompositionof several modules in view point of the Krull-Remak-Schmidt Theorem. Moreover, we generalize the Orzech Theorem and obtain a generalized Vasconcelos Theo-rem. We also discuss the Fitting Lemma of w-modules and prove that the SchurLemma is always correct for w-simple modules. Furthermore, we prove that w-semisimple modules, GV -torsion-free injective modules over a w-Noetherian ringand w-modules with w-composition series can be decomposed into a direct sumof directly indecomposable submodules with local endomorphism rings.
In this paper, we investigate some properties of the ascending and descend-ing chain conditions of w-modules and introduce several new modules, so thatsome classical theories have new representation and application. Firstly, we provethat if R is a w-Noetherian ring, dual modules of finite type modules are finitelypresented type, and discuss the primary decomposition of w-submodules of w-Noetherian modules. We also study the annihilator of a non-zero element of finitetype w-modules. And it is proved that if R is a w-Noetherian ring and M is aGV -torsion-free R-module of finite type, then there exist only a finite number ofmaximal prime ideals, each of which is the annihilator of a non-zero element ofM. Secondly, we show the concept of w-simple modules. And we not only showthe existence of w-simple modules, but also point out the di?erence between w-simple modules and simple modules. Then the definition of the w-semisimplemodule is given. We also obtain new characterizations of semisimple rings andprove that R is semisimple if and only if every w-module is w-semisimple. Andwe discuss the equivalent characterizations of chain conditions of w-submodulesof w-semisimple modules. Meanwhile, we define the w-socle of a w-module, showthat the w-socle and the socle are di?erent and obtain equivalent conditions ofw-Artin modules by means of the w-socle. In addition, we show the concepts ofthe w-Jacobson radical of a w-module and the w-super?uous submodule. We alsogive an example to illustrate that in comparison to the Jacobson radical, the def-inition of the w-Jacobson radical is nontrivial and prove that w(R) ? J(R). Andwe obtain the Nakayama’s Lemma with respect to the w-Jacobson radical. Asthe application, the Kertese Theorem of w-modules is proved. By the definitionof w-addition complements, we character the descending chain conditions of w-modules. At last, we investigate the uniqueness problem of direct decompositionof several modules in view point of the Krull-Remak-Schmidt Theorem. Moreover, we generalize the Orzech Theorem and obtain a generalized Vasconcelos Theo-rem. We also discuss the Fitting Lemma of w-modules and prove that the SchurLemma is always correct for w-simple modules. Furthermore, we prove that w-semisimple modules, GV -torsion-free injective modules over a w-Noetherian ringand w-modules with w-composition series can be decomposed into a direct sumof directly indecomposable submodules with local endomorphism rings.
引文
[1] WANG F G, McCasland R L. On w-modules over strong Mori domains.Comm. Algebra, 1997, 25: 1285-1306.
[2] R. Gilmer. Multiplicative Ideal Theory. Dekker, New York, 1972.
[3] J.R. Hedstrom, E.G. Houston. Some remarks on Star-operations. J. PureAppl. Algebra, 1980, 18: 37-44.
[4] D.D. Anderson, D.F. Anderson. Some remarks on star operations and theclass group. J. Pure Appl. Algebra, 1988, 51: 27-33.
[5]王芳贵.交换环与星型算子理论.北京:科学出版社, 2006.
[6] YIN H Y, WANG F G, ZHU X S, CHEN Y H. w-Modules over commutativerings. to appear.
[7] WANG F G, ZHANG J. Injective modules over w-Noetherian rings. to appear.
[8] WANG F G. On w-coherent domains. to appear.
[9] WANG F G. w-Projective over commutative rings. to appear.
[10]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.
[11]王芳贵,张俊. w-半单性.待发.
[12] Orzech M. Onto endomorphisms are isomorphisms. Amer. Math. Monthly,1971, 78: 357-362.
[13] Kang B G. Integral closure of rings with zero divisors. J. Algebra, 2000, 146:283-290.
[14] Kaplansky I. Commutative Rings (Revised ed.). Chicago: University ofChicago Press, 1974.
[15]陈晋健,陈顺卿.模论.开封:河南大学出版社, 1994.
[16] F.Kasch. Modules and Rings. Academic Press, 1982.
[17] L. A. Kurdachenko, J. Otal, I. Ya. Subbotin. Artinian modules over grouprings. Birkh¨auser, Basel 2007.
[18] WANG F G, McCasland R L. On strong Mori domains. J. Pure Appl. Al-gebra, 1999, 135: 155-165.
[19] F. W. Anderson, K. R. Fuller. Rings and categories of modules. GraduateTexts in Mathematics 13, Springer-Verlag, 1974.
[20] N. Jacobson. Basic Algebra II. San Francisco: W.H. Freeman, 1980.
[21] R. C. Shock. Dual generalizations of the Artinian and Noetherian conditions.Pacific J. Math., 1974, 54: 227-235.
[22]周伯壎.同调代数.北京:科学出版社, 1988.
[23]冯克勤.交换代数基础.北京:高等教育出版社, 1986.
[24]宋光天.交换代数导引.合肥:中国科学技术大学出版社, 2002.
[25]程福长,易忠.环的同调维数.桂林:广西师范大学出版社, 2000.
[26] Faith C. Algebra II, Ring Theory . New York: Springer-Verlag, 1976.
[27] M.Zafrullah. Ascending chain conditions and star operations. Comm. Alge-bra, 1989, 17: 1523-1533.
[28] V. Barucci, D. F. Dobbs. On chain conditions in integral domains. Canad.Math. Bull., 1984, 27: 351-359.
[29] Houston E G, Lucas T, Viswanathan T. Primary decomposition of divisorialideals in Mori domains. J. Algebra, 1988, 117: 327-342.
[30] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge Univer-sity Press, 1986.
[31] Rotman J J. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[32] WANG F G. w-Dimension of domains. Comm. Algebra, 1999, 27(5): 2267-2276.
[33] WANG F G. w-Dimension of domains II. Comm. Algebra, 2001, 29(6): 2419-2428.
[34] WANG F G. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117-120.
[35] WANG F G. On w-Projective modules and w-?at modules. Algebra Colloq.,1997, 4: 112-120.
[36] Kim H, Kim E S, Park Y S. Injective modules over strong Mori domains.Houston J. Math., 2008, 34: 349-360.
[37] Kim H. Module-theoretic characterizations of t-linkative domains. Comm.Algebra, 2008, 36: 1649-1670.
[38] Baghdadi S E L,Gabelli S. w-divisorial domains. J. Pure Appl. Algebra,2005, 285: 335-355.
[2] R. Gilmer. Multiplicative Ideal Theory. Dekker, New York, 1972.
[3] J.R. Hedstrom, E.G. Houston. Some remarks on Star-operations. J. PureAppl. Algebra, 1980, 18: 37-44.
[4] D.D. Anderson, D.F. Anderson. Some remarks on star operations and theclass group. J. Pure Appl. Algebra, 1988, 51: 27-33.
[5]王芳贵.交换环与星型算子理论.北京:科学出版社, 2006.
[6] YIN H Y, WANG F G, ZHU X S, CHEN Y H. w-Modules over commutativerings. to appear.
[7] WANG F G, ZHANG J. Injective modules over w-Noetherian rings. to appear.
[8] WANG F G. On w-coherent domains. to appear.
[9] WANG F G. w-Projective over commutative rings. to appear.
[10]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.
[11]王芳贵,张俊. w-半单性.待发.
[12] Orzech M. Onto endomorphisms are isomorphisms. Amer. Math. Monthly,1971, 78: 357-362.
[13] Kang B G. Integral closure of rings with zero divisors. J. Algebra, 2000, 146:283-290.
[14] Kaplansky I. Commutative Rings (Revised ed.). Chicago: University ofChicago Press, 1974.
[15]陈晋健,陈顺卿.模论.开封:河南大学出版社, 1994.
[16] F.Kasch. Modules and Rings. Academic Press, 1982.
[17] L. A. Kurdachenko, J. Otal, I. Ya. Subbotin. Artinian modules over grouprings. Birkh¨auser, Basel 2007.
[18] WANG F G, McCasland R L. On strong Mori domains. J. Pure Appl. Al-gebra, 1999, 135: 155-165.
[19] F. W. Anderson, K. R. Fuller. Rings and categories of modules. GraduateTexts in Mathematics 13, Springer-Verlag, 1974.
[20] N. Jacobson. Basic Algebra II. San Francisco: W.H. Freeman, 1980.
[21] R. C. Shock. Dual generalizations of the Artinian and Noetherian conditions.Pacific J. Math., 1974, 54: 227-235.
[22]周伯壎.同调代数.北京:科学出版社, 1988.
[23]冯克勤.交换代数基础.北京:高等教育出版社, 1986.
[24]宋光天.交换代数导引.合肥:中国科学技术大学出版社, 2002.
[25]程福长,易忠.环的同调维数.桂林:广西师范大学出版社, 2000.
[26] Faith C. Algebra II, Ring Theory . New York: Springer-Verlag, 1976.
[27] M.Zafrullah. Ascending chain conditions and star operations. Comm. Alge-bra, 1989, 17: 1523-1533.
[28] V. Barucci, D. F. Dobbs. On chain conditions in integral domains. Canad.Math. Bull., 1984, 27: 351-359.
[29] Houston E G, Lucas T, Viswanathan T. Primary decomposition of divisorialideals in Mori domains. J. Algebra, 1988, 117: 327-342.
[30] H. Matsumura. Commutative Ring Theory. Cambridge: Cambridge Univer-sity Press, 1986.
[31] Rotman J J. An Introduction to Homological Algebra. New York: AcademicPress, 1979.
[32] WANG F G. w-Dimension of domains. Comm. Algebra, 1999, 27(5): 2267-2276.
[33] WANG F G. w-Dimension of domains II. Comm. Algebra, 2001, 29(6): 2419-2428.
[34] WANG F G. w-modules over a PVMD. Proceedings of the InternationalSymposium on Teaching and Applications of Engineering Mathematics, HongKong, 2001, 117-120.
[35] WANG F G. On w-Projective modules and w-?at modules. Algebra Colloq.,1997, 4: 112-120.
[36] Kim H, Kim E S, Park Y S. Injective modules over strong Mori domains.Houston J. Math., 2008, 34: 349-360.
[37] Kim H. Module-theoretic characterizations of t-linkative domains. Comm.Algebra, 2008, 36: 1649-1670.
[38] Baghdadi S E L,Gabelli S. w-divisorial domains. J. Pure Appl. Algebra,2005, 285: 335-355.