余模和模的Gorenstein性质
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摘要
在经典同调代数中,模的投射维数、内射维数和平坦维数是重要且基本的研究对象。作为模的投射维数的概念的推广,Auslander和Bridger于1969年在文献[3]中对双侧Noether环R上的有限生成模定义了G-维数的概念(又见文献[2])。几十年后,Enochs,Jenda和Torrecillas [20,21,23]推广了Auslander和Bridger的思想。他们定义了Gorenstein投射,内射和平坦模,并引入了任意模的Gorenstein同调维数的概念。Avramov, Buchweitz, Martsinkovsky和Reiten证明了,对Noether环上的有限生成模M,M是Gorenstein投射的当且仅当它的Gorenstein投射维数是0[10]。
近年来,这些Gorenstein模已经有了很多形式的推广。特别地,Asensio, Enochs等人([1,24])在余模范畴中定义了Gorenstein内射和投射余模,并且研究了余模的Gorenstein内射和投射维数。而Bennis等人([7])引入了X-Gorenstein投射模的概念,其中X是包含投射模的模类。
本文中,我们在任意的余代数C上定义并研究了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模。同时,我们引入并研究了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。最后,我们研究了Gorenstein平坦(余挠)维数,FP-内射(FP-投射)维数和余挠对在A-Mod和A#H-Mod之间的关系。
全文共分四章,内容如下
第一章是引言,主要介绍了问题的背景和预备知识。
在第二章中,我们在任意的余代数C上定义了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模,并且证明了,在左半完全余代数C上,左C余模M是(弱)Gorenstein余平坦余模当且仅当M是(弱)Gorenstein内射余模。同时,我们研究了弱Gorenstein内射余模预覆盖和预包络的存在性问题。我们证明了在左半完全余代数C上,任意的C余模都有一个弱Gorenstein内射预包络(覆盖)。
第三章我们引入了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。我们证明了关于Gorenstein投射,内射和平坦模的主要结果对于X-Gorenstein投射右R-模,y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模仍然成立。
设H是域k上的有限维Hopf代数,A是左H-模代数。在第四章中,我们证明了在右凝聚环A(或任意环A)上,如果A#H/A可分并且φ:A→A#H是可裂的(A,A)双模单同态,那么l.Gwd(A)=l.Gwd(A#H)(或l.cotD(A)=l.cotD(A#H)和l.Gcd(A)= l.Gcd(A#H))。同时,我们研究了FP-内射(FP-投射)维数在有限维Hopf代数下的作用。我们证明了在任意环A上,如果A#H/A可分并且φ：A→A#H是可裂的(A,A)双模单同态,那么l.FP-dim(A)=l.FP-dim(A#H),并且l f p D(A)=l fpD(A#H).最后,我们给出了余挠对在A-Mod和A#H-Mod之间的关系。
In classical homological algebra, projective, injective and flat dimensions of mod-ules are important and fundamental research objects. As a generalization of the notion of projective dimension of modules, Auslander and Bridger [3] introduced in 1969 the G-dimension, G-dimM, for every finitely generated R-module M (see also [2]) for a two-sided Noetherian ring R. Several decades later, Enochs, Jenda, and Torrecillas [20,21,23] extended the idea of Auslander and Bridger. They defined Gorenstein pro-jective, injective and flat modules, and the related Gorenstein homological dimension for any module M. Avramov, Buchweitz, Martsinkovsky and Reiten proved that a finitely generated module over a Noetherian ring is Gorenstein projective if and only if G-dimRM= 0 [10].
In the recent years, these Gorenstein modules have become a vigorously active area of research. In particular, Asensio, Enochs et al. ([1,24]) defined Gorenstein injective and projective C-comodules in the category of comodules. Along the same lines, Gorenstein injective and projective dimensions of a comodule were also introduced and studied. Let X be a class of right R-modules that contains all projective right R-modules, Bennis et al.([7]) introduced the notion of X-Gorenstein projective right R-modules.
In this dissertation, we introduce and study Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We also introduce and study y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. Finally we study the relationship of Gorenstein flat (cotorsion) dimen-sions, FP-injective (FP-projective) dimensions and cotorsion pairs between A-Mod and A#H-Mod.
This paper consists of four chapters.
In Chapter 1, some backgrounds and preliminaries are given.
Chapter 2 is devoted to introducing Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We prove that, for a left semiperfect coalgebra C, a left C-comodule M is (weakly) Gorenstein coflat if and only if M is (weakly) Gorenstein injective. We also study the existence of pre-covers and preenvelopes by weakly Gorenstein injective comodules. It is shown that every left C-comodule has a weakly Gorenstein injective preenvelope (cover) over a left semiperfect coalgebra C.
Chapter 3 is devoted to studying y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. We show that principal results on Gorenstein projective, injective and flat modules remain true for X-Gorenstein projective right R-modules, y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules..
Let H be a finite-dimensional Hopf algebra over a field k and A a left H-module algebra. In Chapter 4, we prove that over a right coherent ring A (resp. any ring A), if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A, A)-bimodules, l.Gwd(A)= l.Gwd(A#H) (resp. l.cotD(A)=l.cotD(A#H) and l.Gcd(A)= l.Gcd(A#H)). We also study the FP-injective (FP-projective) dimen-sions under actions of finite-dimensional Hopf algebras. We get that if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A,A)-bimodules, then l.FP-dim(A)=l.FP-dim(A#H) and l f pD(A)=lfpD(A#H). Finally, we give a con-nection of cotorsion pairs between A-Mod and A#H-Mod.
近年来,这些Gorenstein模已经有了很多形式的推广。特别地,Asensio, Enochs等人([1,24])在余模范畴中定义了Gorenstein内射和投射余模,并且研究了余模的Gorenstein内射和投射维数。而Bennis等人([7])引入了X-Gorenstein投射模的概念,其中X是包含投射模的模类。
本文中,我们在任意的余代数C上定义并研究了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模。同时,我们引入并研究了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。最后,我们研究了Gorenstein平坦(余挠)维数,FP-内射(FP-投射)维数和余挠对在A-Mod和A#H-Mod之间的关系。
全文共分四章,内容如下
第一章是引言,主要介绍了问题的背景和预备知识。
在第二章中,我们在任意的余代数C上定义了Gorenstein余平坦和弱Gorenstein内射(余平坦)余模,并且证明了,在左半完全余代数C上,左C余模M是(弱)Gorenstein余平坦余模当且仅当M是(弱)Gorenstein内射余模。同时,我们研究了弱Gorenstein内射余模预覆盖和预包络的存在性问题。我们证明了在左半完全余代数C上,任意的C余模都有一个弱Gorenstein内射预包络(覆盖)。
第三章我们引入了y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模,其中y是包含所有内射右R-模的右R-模类。我们证明了关于Gorenstein投射,内射和平坦模的主要结果对于X-Gorenstein投射右R-模,y-Gorenstein内射右R-模和y-Gorenstein平坦左R-模仍然成立。
设H是域k上的有限维Hopf代数,A是左H-模代数。在第四章中,我们证明了在右凝聚环A(或任意环A)上,如果A#H/A可分并且φ:A→A#H是可裂的(A,A)双模单同态,那么l.Gwd(A)=l.Gwd(A#H)(或l.cotD(A)=l.cotD(A#H)和l.Gcd(A)= l.Gcd(A#H))。同时,我们研究了FP-内射(FP-投射)维数在有限维Hopf代数下的作用。我们证明了在任意环A上,如果A#H/A可分并且φ：A→A#H是可裂的(A,A)双模单同态,那么l.FP-dim(A)=l.FP-dim(A#H),并且l f p D(A)=l fpD(A#H).最后,我们给出了余挠对在A-Mod和A#H-Mod之间的关系。
In classical homological algebra, projective, injective and flat dimensions of mod-ules are important and fundamental research objects. As a generalization of the notion of projective dimension of modules, Auslander and Bridger [3] introduced in 1969 the G-dimension, G-dimM, for every finitely generated R-module M (see also [2]) for a two-sided Noetherian ring R. Several decades later, Enochs, Jenda, and Torrecillas [20,21,23] extended the idea of Auslander and Bridger. They defined Gorenstein pro-jective, injective and flat modules, and the related Gorenstein homological dimension for any module M. Avramov, Buchweitz, Martsinkovsky and Reiten proved that a finitely generated module over a Noetherian ring is Gorenstein projective if and only if G-dimRM= 0 [10].
In the recent years, these Gorenstein modules have become a vigorously active area of research. In particular, Asensio, Enochs et al. ([1,24]) defined Gorenstein injective and projective C-comodules in the category of comodules. Along the same lines, Gorenstein injective and projective dimensions of a comodule were also introduced and studied. Let X be a class of right R-modules that contains all projective right R-modules, Bennis et al.([7]) introduced the notion of X-Gorenstein projective right R-modules.
In this dissertation, we introduce and study Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We also introduce and study y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. Finally we study the relationship of Gorenstein flat (cotorsion) dimen-sions, FP-injective (FP-projective) dimensions and cotorsion pairs between A-Mod and A#H-Mod.
This paper consists of four chapters.
In Chapter 1, some backgrounds and preliminaries are given.
Chapter 2 is devoted to introducing Gorenstein coflat comodules and weakly Gorenstein injective (resp. coflat) comodules over any coalgebra C. We prove that, for a left semiperfect coalgebra C, a left C-comodule M is (weakly) Gorenstein coflat if and only if M is (weakly) Gorenstein injective. We also study the existence of pre-covers and preenvelopes by weakly Gorenstein injective comodules. It is shown that every left C-comodule has a weakly Gorenstein injective preenvelope (cover) over a left semiperfect coalgebra C.
Chapter 3 is devoted to studying y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules, where y is a class of right R-modules that contains all injective right R-modules. We show that principal results on Gorenstein projective, injective and flat modules remain true for X-Gorenstein projective right R-modules, y-Gorenstein injective right R-modules and y-Gorenstein flat left R-modules..
Let H be a finite-dimensional Hopf algebra over a field k and A a left H-module algebra. In Chapter 4, we prove that over a right coherent ring A (resp. any ring A), if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A, A)-bimodules, l.Gwd(A)= l.Gwd(A#H) (resp. l.cotD(A)=l.cotD(A#H) and l.Gcd(A)= l.Gcd(A#H)). We also study the FP-injective (FP-projective) dimen-sions under actions of finite-dimensional Hopf algebras. We get that if A#H/A is separable andφ:A→A#H is a splitting monomorphism of (A,A)-bimodules, then l.FP-dim(A)=l.FP-dim(A#H) and l f pD(A)=lfpD(A#H). Finally, we give a con-nection of cotorsion pairs between A-Mod and A#H-Mod.
引文
[1]Asensio, M. J., Lopez Ramos J. A. and Torrecillas, B. Gorenstein coalgebras, Acta Math. Hungar.,1999,85:187-198.
[2]Auslander, M. Anneaux de Gorenstein et torsion en algebre commutative, Seminaire d'algebre commutative 1966/67, notes by Mangeney, M., Peskine C. and Szpiro, L. Ecole Normale Superieure de Jeunes Filles, Paris, Secretariat mathematique,1967. MR 37:1435.
[3]Auslander, M. and Bridger, M. Stable Module Theory. Memoirs Amer Math Soc, Vol 94. Providence:Amer Math Soc,1969.
[4]Auslander, M., Reiten, I. and Smalφ, S. O. Representation Theory of Artin Algebras, Cambridge University Press, Cambridge,1997.
[5]Bennis, D. Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra,2009,37 (3):855-868.
[6]Bennis, D. and Mahdou, N. Global Gorenstein dimensions, Proc. Amer. Math. Soc., 2010,138 (2):461-465.
[7]Bennis, D. and Ouarghi, K. χ-Gorenstein projective modules, International Mathe-matical Forum,2010,5 (10):487-491.
[8]Brzezinski, T. and Wisbauer, R. Corings and Comodules, London Math. Soc. Lecture Note Ser.,309. Cambridge Univ. Press, Cambridge,2003.
[9]Castao Iglesias, F., Gomez-Torrecillas, J. and Nastasescu, C. Frobenius functors:ap-plications. Comm. Algebra,1999,27 (10):4879-4900.
[10]Christensen, L. W. Gorenstein Dimensions, Lecture Notes in Math.1747, Springer, Berlin, Heidelberg,2000.
[11]Dascalescu, S., Nastasescu C. and Raianu, S. Hopf Algebras:An Introduction, Monogr. Textbooks Pure Appl. Math.235. Marcel Dekker, New York.2001.
[12]Ding, N. Q., Li, Y. L. and Mao, L. X. Strongly Gorenstein flat modules, J. Aust. Math. Soc.,2009,86:323-338.
[13]Doi, Y. Homological coalgebra, J. Math. Soc. Japan,1981,33:31-50.
[14]Doi, Y. Hopf extensions of algebras and Maschke type theorems, Israel J. Math.,1990, 72:99-108.
[15]El Bashir, R. Covers and directed Colimits, Alge.br. Represent. Theory,2006,9:423-430.
[16]Enochs, E. E. Flat covers and flat cotorsion modules. Proc. Amer. Math. Soc.,1984, 92 (2):179-184.
[17]Enochs, E. E. Injective and flat covers, envelopes and resolvents, Israel J. Math.,1981, 39:189-209.
[18]Enochs, E. E. and Jenda, O. M. G. Relative Homological Algebra, de Gruyter Exp. Math.30, de Gruyter, Berlin,2000.
[19]Enochs, E. E. and Jenda, O. M. G. Copure injective resolutions, flat Resolvents and dimensions, Comment. Math. Univ. Carolin,1993,34 (2):203-211.
[20]Enochs, E. E. and Jenda, O. M. G. Gorenstein injective and Gorenstein projective modules, Math. Z.,1995,220:611-633.
[21]Enochs, E. E. and Jenda, O. M. G. Gorenstein flat preenvelopes and resolvents. Nan-jing Daxue Xuebao Shuxue Bannian Kan,1995,220:611-633.
[22]Enochs, E. E., Jenda, O. M. G. and Lopez-Ramos, J. A. The existence of Gorenstein flat covers, Math. Scand.,2004,94:46-62.
[23]Enochs, E. E., Jenda, O. M. G. and Torrecillas, B. Gorenstein flat modules. J Nanjing Univ Math Biquarterly,1993,10 (1):1-9.
[24]Enochs, E. E. and Lopez-Ramos, J. A. Relative homological coalgebra, Acta Math. Hungar.,2004,104 (4):331-343.
[25]Enochs, E. E. and Lopez-Ramos, J. A. Gorenstein Flat Modules, Huntington, NY. Nova Science Publishers, Inc.2001.
[26]Enochs, E. E. and Oyonarte, L. Covers, Envelopes and Cotorsion Theories, Nova Science Publishers, Inc, New York,2002.
[27]Fieldhouse, D. J. Character modules, dimension and purity, Glasgow Math. J.,1972, 13:144-146.
[28]Garcia Rozas, J. R. and Torrecillas, B. Preserving and reflecting covers by functors. Applications to graded modules. J. Pure Appl. Algebra,1996,112:91-107.
[29]Garcia Rozas, J. R. and Torrecillas, B. Preserving of covers by Hopf invariants, Comm. Algebra,1999,27 (4):1737-1750.
[30]Gobel, R. and Trlifaj, J. Approximations and Endomorphism Algebras of Modules, GEM 41, Walter de Gruyter, Berlin-New York,2006.
[31]Holm, H. Gorenstein homological dimensions, J. Pure Appl. Algebra,2004,189:167-193.
[32]Huang, Z. Y. and Sun, J. X. Tilting modules for the crossed products, Preprint.
[33]Lam, T. Y. Lectures on Modules and Rings, New York-Heidelberg-Berlin, Springer-Verlag 1999.
[34]Lin, B. I.-p. Semiperfect coalgebras, J. Algebra,1977,49:357-373.
[35]Lopez-Ramos, J. A. Gorenstein injective and projective modules and actions of finite-dimensional Hopf algebras, Ark. Mat.,2008,46:349-361.
[36]Mao, L. X. and Ding, N. Q. Gorenstein FP-injective and Gorenstein flat modules, J. Algebra Appl,2008,7:491-506.
[37]Mao L. X. and Ding, N. Q. FP-projective dimensions, Comm. Algebra,2005,33: 1153-1170.
[38]Mao L. X. and Ding, N. Q. The cotorsion dimension of modules and rings, Abelian groups, rings, modules, and homological algebra,217-233, Lect. Notes Pure Appl. Math.249,2006.
[39]Megibben, C. Absolutely pure modules, Proc. Amer. Math. Soc.,1970,26 (4):561-566.
[40]Montgomery, S. Hopf Algebras and Their Actions on Rings, CBMS Regional Confer-ence Series in Mathematics 82, Amer. Math. Soc. Providence, RI 1993.
[41]Nastasescu, C., Torrecillas B. and Zhang, Y. H. Hereditary coalgebras, Comm. Alge-bra,1996,24:1521-1528.
[42]Nastasescu C., Van den Bergh M. and Van Oystaeyen, F. Separable functors applied to graded rings, J. Algebra,1989,123:397-413.
[43]Rotman, J. J. An Introduction to Homological Algebra, Academic Press, New York, 1979.
[44]Stenstrom, B. Ring of Quotients:An Introduction to Methods of Ring Theory, New York-Berlin. Springer-Verlag,1975.
[45]Stenstrom, B. Direct sum decomposition in Grothendieck categories, Ark. Mat.,1968, 7:427-432.
[46]Stenstrom, B. Coherent rings and FP-injective modules, J. London Math. Soc.,1970, 2:323-329.
[47]Tamekkante, M. The right orthogonal class gP(R)⊥ via Ext, arXiv:0911.1272.
[48]Van Oystaeyen, F., Xu, Y. and Zhang, Y. Inductions and coinductions for Hopf extensions, Sci. China Ser. A,1996,39:246-263.
[2]Auslander, M. Anneaux de Gorenstein et torsion en algebre commutative, Seminaire d'algebre commutative 1966/67, notes by Mangeney, M., Peskine C. and Szpiro, L. Ecole Normale Superieure de Jeunes Filles, Paris, Secretariat mathematique,1967. MR 37:1435.
[3]Auslander, M. and Bridger, M. Stable Module Theory. Memoirs Amer Math Soc, Vol 94. Providence:Amer Math Soc,1969.
[4]Auslander, M., Reiten, I. and Smalφ, S. O. Representation Theory of Artin Algebras, Cambridge University Press, Cambridge,1997.
[5]Bennis, D. Rings over which the class of Gorenstein flat modules is closed under extensions, Comm. Algebra,2009,37 (3):855-868.
[6]Bennis, D. and Mahdou, N. Global Gorenstein dimensions, Proc. Amer. Math. Soc., 2010,138 (2):461-465.
[7]Bennis, D. and Ouarghi, K. χ-Gorenstein projective modules, International Mathe-matical Forum,2010,5 (10):487-491.
[8]Brzezinski, T. and Wisbauer, R. Corings and Comodules, London Math. Soc. Lecture Note Ser.,309. Cambridge Univ. Press, Cambridge,2003.
[9]Castao Iglesias, F., Gomez-Torrecillas, J. and Nastasescu, C. Frobenius functors:ap-plications. Comm. Algebra,1999,27 (10):4879-4900.
[10]Christensen, L. W. Gorenstein Dimensions, Lecture Notes in Math.1747, Springer, Berlin, Heidelberg,2000.
[11]Dascalescu, S., Nastasescu C. and Raianu, S. Hopf Algebras:An Introduction, Monogr. Textbooks Pure Appl. Math.235. Marcel Dekker, New York.2001.
[12]Ding, N. Q., Li, Y. L. and Mao, L. X. Strongly Gorenstein flat modules, J. Aust. Math. Soc.,2009,86:323-338.
[13]Doi, Y. Homological coalgebra, J. Math. Soc. Japan,1981,33:31-50.
[14]Doi, Y. Hopf extensions of algebras and Maschke type theorems, Israel J. Math.,1990, 72:99-108.
[15]El Bashir, R. Covers and directed Colimits, Alge.br. Represent. Theory,2006,9:423-430.
[16]Enochs, E. E. Flat covers and flat cotorsion modules. Proc. Amer. Math. Soc.,1984, 92 (2):179-184.
[17]Enochs, E. E. Injective and flat covers, envelopes and resolvents, Israel J. Math.,1981, 39:189-209.
[18]Enochs, E. E. and Jenda, O. M. G. Relative Homological Algebra, de Gruyter Exp. Math.30, de Gruyter, Berlin,2000.
[19]Enochs, E. E. and Jenda, O. M. G. Copure injective resolutions, flat Resolvents and dimensions, Comment. Math. Univ. Carolin,1993,34 (2):203-211.
[20]Enochs, E. E. and Jenda, O. M. G. Gorenstein injective and Gorenstein projective modules, Math. Z.,1995,220:611-633.
[21]Enochs, E. E. and Jenda, O. M. G. Gorenstein flat preenvelopes and resolvents. Nan-jing Daxue Xuebao Shuxue Bannian Kan,1995,220:611-633.
[22]Enochs, E. E., Jenda, O. M. G. and Lopez-Ramos, J. A. The existence of Gorenstein flat covers, Math. Scand.,2004,94:46-62.
[23]Enochs, E. E., Jenda, O. M. G. and Torrecillas, B. Gorenstein flat modules. J Nanjing Univ Math Biquarterly,1993,10 (1):1-9.
[24]Enochs, E. E. and Lopez-Ramos, J. A. Relative homological coalgebra, Acta Math. Hungar.,2004,104 (4):331-343.
[25]Enochs, E. E. and Lopez-Ramos, J. A. Gorenstein Flat Modules, Huntington, NY. Nova Science Publishers, Inc.2001.
[26]Enochs, E. E. and Oyonarte, L. Covers, Envelopes and Cotorsion Theories, Nova Science Publishers, Inc, New York,2002.
[27]Fieldhouse, D. J. Character modules, dimension and purity, Glasgow Math. J.,1972, 13:144-146.
[28]Garcia Rozas, J. R. and Torrecillas, B. Preserving and reflecting covers by functors. Applications to graded modules. J. Pure Appl. Algebra,1996,112:91-107.
[29]Garcia Rozas, J. R. and Torrecillas, B. Preserving of covers by Hopf invariants, Comm. Algebra,1999,27 (4):1737-1750.
[30]Gobel, R. and Trlifaj, J. Approximations and Endomorphism Algebras of Modules, GEM 41, Walter de Gruyter, Berlin-New York,2006.
[31]Holm, H. Gorenstein homological dimensions, J. Pure Appl. Algebra,2004,189:167-193.
[32]Huang, Z. Y. and Sun, J. X. Tilting modules for the crossed products, Preprint.
[33]Lam, T. Y. Lectures on Modules and Rings, New York-Heidelberg-Berlin, Springer-Verlag 1999.
[34]Lin, B. I.-p. Semiperfect coalgebras, J. Algebra,1977,49:357-373.
[35]Lopez-Ramos, J. A. Gorenstein injective and projective modules and actions of finite-dimensional Hopf algebras, Ark. Mat.,2008,46:349-361.
[36]Mao, L. X. and Ding, N. Q. Gorenstein FP-injective and Gorenstein flat modules, J. Algebra Appl,2008,7:491-506.
[37]Mao L. X. and Ding, N. Q. FP-projective dimensions, Comm. Algebra,2005,33: 1153-1170.
[38]Mao L. X. and Ding, N. Q. The cotorsion dimension of modules and rings, Abelian groups, rings, modules, and homological algebra,217-233, Lect. Notes Pure Appl. Math.249,2006.
[39]Megibben, C. Absolutely pure modules, Proc. Amer. Math. Soc.,1970,26 (4):561-566.
[40]Montgomery, S. Hopf Algebras and Their Actions on Rings, CBMS Regional Confer-ence Series in Mathematics 82, Amer. Math. Soc. Providence, RI 1993.
[41]Nastasescu, C., Torrecillas B. and Zhang, Y. H. Hereditary coalgebras, Comm. Alge-bra,1996,24:1521-1528.
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