PI-内射模与强Copure-内射模
|
摘要
本学位论文共分为三章：
第一章引入了PI-内射模与PI-投射模,证明了PI-投射模类与PI-内射模类构成一个完备的余挠对,借助这些我们定义了弱完全环并给出了Noether环、von Neumann正则环和半单环的一些新刻画.
第二章首先证明了第一章构造的余挠对是遗传的当且仅当环R的每个纯右理想是PI-投射的.进一步地,在每个纯右理想都是PI-投射模的环R上,定义了PI-投射维数与PI-内射维数.我们发现PI-投射维数与PI-内射维数能分别度量一个环与von Neumann正则环与弱完全环之间的差距.
第三章我们证明了在交换Artin环上,每个R-模都有强Copure-内射复盖.同时我们利用Hom的左导出函子及R-模的左强Copure-内射分解刻画了模与环的Copure-内射维数.
This thesis is made up of three chapters:
In Chapter 1, the notions of PI-injective module and PI-projective module are introduced. With the help of these objects, we prove that the class of PI-injective mod-ules and the class of PI-projective modules can be constructed to be a completely cotor-sion pair. Furthermore, we define the weakly perfect rings and give some new charac-terizations for some well-known rings, such as von Neumann regular rings, noetherian rings and semisimple Artinian rings.
In Chapter 2, the cotorsion pair constructed in Chapter 1 is hereditary if and only if every pure right ideal I of R is PI-projective is proved at first. Furthermore, PI-projective dimension and PI-injective dimension for modules and rings are defined respectively and it turns out that PI-projective dimension and PI-injective dimension measure how far away a ring is from being von Neumann regular and weakly perfect respectively provided that every pure right ideal of R is PI-projective.
In Chapter 3, the existence of the strongly copure injective cover is proved over a commutative artinian ring at first. Furthermore, we discuss the copure injective dimen-sions of modules and rings in terms of the left derived functor of Hom.
第一章引入了PI-内射模与PI-投射模,证明了PI-投射模类与PI-内射模类构成一个完备的余挠对,借助这些我们定义了弱完全环并给出了Noether环、von Neumann正则环和半单环的一些新刻画.
第二章首先证明了第一章构造的余挠对是遗传的当且仅当环R的每个纯右理想是PI-投射的.进一步地,在每个纯右理想都是PI-投射模的环R上,定义了PI-投射维数与PI-内射维数.我们发现PI-投射维数与PI-内射维数能分别度量一个环与von Neumann正则环与弱完全环之间的差距.
第三章我们证明了在交换Artin环上,每个R-模都有强Copure-内射复盖.同时我们利用Hom的左导出函子及R-模的左强Copure-内射分解刻画了模与环的Copure-内射维数.
This thesis is made up of three chapters:
In Chapter 1, the notions of PI-injective module and PI-projective module are introduced. With the help of these objects, we prove that the class of PI-injective mod-ules and the class of PI-projective modules can be constructed to be a completely cotor-sion pair. Furthermore, we define the weakly perfect rings and give some new charac-terizations for some well-known rings, such as von Neumann regular rings, noetherian rings and semisimple Artinian rings.
In Chapter 2, the cotorsion pair constructed in Chapter 1 is hereditary if and only if every pure right ideal I of R is PI-projective is proved at first. Furthermore, PI-projective dimension and PI-injective dimension for modules and rings are defined respectively and it turns out that PI-projective dimension and PI-injective dimension measure how far away a ring is from being von Neumann regular and weakly perfect respectively provided that every pure right ideal of R is PI-projective.
In Chapter 3, the existence of the strongly copure injective cover is proved over a commutative artinian ring at first. Furthermore, we discuss the copure injective dimen-sions of modules and rings in terms of the left derived functor of Hom.
引文
[1]R. B. Warfield. Purity and algebraic compactness for modules[J]. Pacific J. Math., 1969,28:699-719.
[2]E. E. Enochs. Flat covers and flat cotorsion modules[J]. Proc. Amer. Math. Soc., 1984,92 (2):179-184.
[3]毛立新.包络复盖理论及其应用[D].南京：南京大学数学系,2005.
[4]L. X. Mao, N. Q. Ding. Notes on cotorsion modules[J]. Comm. Algebra,2005,33 (1):349-360.
[5]L. X. Mao, N. Q. Ding. Cotorsion modules and relative pure-injectivity[J]. J. Aust. Math. Soc.,2006,81:225-243.
[6]L. Bican, E. Bashir, E. E. Enochs. All modules has flat covers[J]. Bull. London Math. Soc.,2001,33:385-490.
[7]J. Rada, M. Saorin. Rings characterized by (pre)envelopes and (pre)covers of their moduIes[J]. Comm. Algebra,1998,26 (3):899-912.
[8]E. E. Enochs, O. M. G. Jenda. Relative Homological Algebra[M]. Berlin-New York:Walter de Gruyter,2000.
[9]H. Holm, P. J(?)rgensen. Covers, precovers, and purity[J]. Illinois J. Math.,2008, 52:719-735.
[10]E. E. Enochs. Injective and flat covers, envelopes and resolvents [J]. Israel J. Math., 1981,39:189-209.
[11]N. Q. Ding. On envelopes with the unique mapping property [J]. Comm. Algebra, 1996,24 (4):1459-1470.
[12]E. E. Enochs, O. M. G. Jenda, J. A. Lopez-Ramos. The existence of Gorenstein flat covers[J]. Math. Scand.,2004,94:46-62.
[13]B. Madox. Absolutely pure modules[J]. Proc. Amer. Math. Soc.,1967,18:155-158.
[14]B. Stenstrom. Coherent rings and FP-injective modules[J]. J. London Math. Soc., 1970,2:323-329.
[15]F. W. Anderson, K. R. Fuller. Rings and Categories of Modules[M]. New York: Springer-Verlag,1974.
[16]J. Xu. Flat covers of Modules[M]. Lecture Notes in Mathematics 1634, Springer-Verlag:Berlin-Heidelberg-New York,1996.
[17]P. F. Smith. Injective modules and prime ideals[J]. Comm. Algebra,1981,9:989-999.
[18]J. Dauns. Modules and Rings[M]. Cambridge:Cambridge University Press,1994.
[19]T. Y. Lam. Lectures on Modules and Rings[M]. New York:Springer-Verlag,1999.
[20]E. E. Enochs. A note on absolutely pure modules[J]. Canada. Math. Bull.,1976, 19:361-362.
[21]E. E. Enochs, O. M. G. Jenda. Copure injective modules [J]. Quaest. Math.,1991, 14:401-409.
[22]E. E. Enochs, O. M. G. Jenda. Copure injective resolutions, flat resolvents and dimensions[J]. Comment. Math.,1993,34,203-211.
[23]L. X. Mao, N. Q. Ding. Relative copure injective and copure flat modules [J]. J. Pure Appl. Algebra,2007,208:635-646.
[24]L. X. Mao, N. Q. Ding. FI-injective and FI-flat modules[J]. J. Algebra,2007, 309:367-385.
[25]L. X. Mao, N. Q. Ding. Relative FP-projective modules[J]. Comm. Algebra, 2005,33 (5):1587-1602.
[26]J. J. Rotman. An Introduction to Homological Algebra[M]. New York:Academic Piess,1979.
[2]E. E. Enochs. Flat covers and flat cotorsion modules[J]. Proc. Amer. Math. Soc., 1984,92 (2):179-184.
[3]毛立新.包络复盖理论及其应用[D].南京：南京大学数学系,2005.
[4]L. X. Mao, N. Q. Ding. Notes on cotorsion modules[J]. Comm. Algebra,2005,33 (1):349-360.
[5]L. X. Mao, N. Q. Ding. Cotorsion modules and relative pure-injectivity[J]. J. Aust. Math. Soc.,2006,81:225-243.
[6]L. Bican, E. Bashir, E. E. Enochs. All modules has flat covers[J]. Bull. London Math. Soc.,2001,33:385-490.
[7]J. Rada, M. Saorin. Rings characterized by (pre)envelopes and (pre)covers of their moduIes[J]. Comm. Algebra,1998,26 (3):899-912.
[8]E. E. Enochs, O. M. G. Jenda. Relative Homological Algebra[M]. Berlin-New York:Walter de Gruyter,2000.
[9]H. Holm, P. J(?)rgensen. Covers, precovers, and purity[J]. Illinois J. Math.,2008, 52:719-735.
[10]E. E. Enochs. Injective and flat covers, envelopes and resolvents [J]. Israel J. Math., 1981,39:189-209.
[11]N. Q. Ding. On envelopes with the unique mapping property [J]. Comm. Algebra, 1996,24 (4):1459-1470.
[12]E. E. Enochs, O. M. G. Jenda, J. A. Lopez-Ramos. The existence of Gorenstein flat covers[J]. Math. Scand.,2004,94:46-62.
[13]B. Madox. Absolutely pure modules[J]. Proc. Amer. Math. Soc.,1967,18:155-158.
[14]B. Stenstrom. Coherent rings and FP-injective modules[J]. J. London Math. Soc., 1970,2:323-329.
[15]F. W. Anderson, K. R. Fuller. Rings and Categories of Modules[M]. New York: Springer-Verlag,1974.
[16]J. Xu. Flat covers of Modules[M]. Lecture Notes in Mathematics 1634, Springer-Verlag:Berlin-Heidelberg-New York,1996.
[17]P. F. Smith. Injective modules and prime ideals[J]. Comm. Algebra,1981,9:989-999.
[18]J. Dauns. Modules and Rings[M]. Cambridge:Cambridge University Press,1994.
[19]T. Y. Lam. Lectures on Modules and Rings[M]. New York:Springer-Verlag,1999.
[20]E. E. Enochs. A note on absolutely pure modules[J]. Canada. Math. Bull.,1976, 19:361-362.
[21]E. E. Enochs, O. M. G. Jenda. Copure injective modules [J]. Quaest. Math.,1991, 14:401-409.
[22]E. E. Enochs, O. M. G. Jenda. Copure injective resolutions, flat resolvents and dimensions[J]. Comment. Math.,1993,34,203-211.
[23]L. X. Mao, N. Q. Ding. Relative copure injective and copure flat modules [J]. J. Pure Appl. Algebra,2007,208:635-646.
[24]L. X. Mao, N. Q. Ding. FI-injective and FI-flat modules[J]. J. Algebra,2007, 309:367-385.
[25]L. X. Mao, N. Q. Ding. Relative FP-projective modules[J]. Comm. Algebra, 2005,33 (5):1587-1602.
[26]J. J. Rotman. An Introduction to Homological Algebra[M]. New York:Academic Piess,1979.