Gorenstein投射模何时是Ding投射模（英文）
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摘要
本文主要利用强Gorenstein投射模、相对纯投射模等概念,研究了何时每个Gorenstein投射模是Ding投射模.作为应用,我们证明了:若R是1-FC环,则每个Gorenstein投射左或右R-模均是Ding投射模.
We characterize when every Gorenstein projective module is Ding projective in terms of strongly Gorenstein projective modules, relative pure projective modules. As a corollary,we prove that any Gorenstein projective left or right R-module is Ding projective provided that R is a 1-FC ring.
We characterize when every Gorenstein projective module is Ding projective in terms of strongly Gorenstein projective modules, relative pure projective modules. As a corollary,we prove that any Gorenstein projective left or right R-module is Ding projective provided that R is a 1-FC ring.
引文
[1] Auslander M. Anneaux de Gorenstein, et torsion en algèbre commutative. S6minaire d'Algèbre Commutative dirige par Pierre Samuel(1966/67), Secretariat mathématique, Paris, 1967.
[2] Auslander M and Bridger M. Stable Module Theory. Memoirs. Amer. Math. Soc., Vol. 94,American Mathematical Society, Providence, RI, 1969.
[3] Bennis D. A note on Gorenstein flat dimension. Algebra Colloq., 2011, 18:155-161.
[4] Bennis D and Mahdou N. A Generalization of Strongly Gorenstein Projective Modules. J. Algebra Appl., 2009, 8:219-227.
[5] Bennis D and Mahdou N. Global Gorenstein Dimensions. Proc. Amer. Math. Soc., 2010, 138:461-465.
[6] Bennis D and Mahdou N. Strongly Gorenstein Projective, Injective, and Flat Modules. J. Pure Appl. Algebra, 2007, 210:437-445.
[7] Christensen L W. Gorenstein Dimensions. Lecture Notes in Math., Vol. 1747, Springer, Berlin,2000.
[8] Ding N Q and Chen J L. Coherent Rings with Finite Self-FP-injective Dimension. Comm. Algebra,1996, 24(9):2963-2980.
[9] Ding N Q and Chen J L. The Flat Dimensions of Injective Modules. Manuscripta Math., 1993, 78:165-177.
[10] Ding N Q, Li Y L and Mao L X. Strongly Gorenstein Flat Modules. J. Aust. Math. Soc., 2009, 86:323-338.
[11] Ding N Q and Mao L X. Relative FP-projective Modules. Comm. Algebra, 2005, 33:1587-1602.
[12] Enochs E E and Jenda O M G. Gorenstein Injective and Projective Modules. Math. Z., 1995, 220:611-633.
[13] Enochs E E and Jenda O M G. Relative Homological Algebra. Walter de Gruyter, Berlin, New York, 2000.
[14] Enochs E E, Jenda O M G and Torrecillas B. Gorenstein Flat Modules. Nanjing Daxue Xuebao Shuxue Bannian Kan,1993, 10:1-9.
[15] Fieldhouse D J. Character Modules, Dimension and Purity. Glasgow Math. J., 1972, 13:144-146.
[16] Gillespie J. Model Structures on Modules over Ding-Chen Rings. Homology, Homotopy Appl., 2010,12:61-73.
[17] Gillespie J. On Ding Injective, Ding Projective, and Ding Flat Modules and Complexes.ArXiv:1512.05999v1.
[18] G?bel R and Trlifaj J. Approximations and Endomorphism Algebras of Modules. Walter de Gruyter,Berlin, New York, 2006.
[19] Holm H. Gorenstein Homological Dimensions. J. Pure. Appl. Algebra, 2004, 189:167-193. Proc.Amer. Math. Soc. 1967, 18:155-158.
[20] Mahdou N and Tamekkante M. Strongly Gorenstein Flat Modules and Dimensions. Chin. Ann.Math., 2011, 32B(4):533-548.
[21] Stenstr?m B. Coherent Rings and FP-injective Modules. J. London Math. Soc., 1970, 2:323-329.
[22] Wang J, Li Y X and Hu J S. Noncommutative G-semihereditary Rings. J. Algebra Appl., 2018, 16:1850014, https://doi.org/10.1142/S0219498818500147.
[23] Warfield R R. Purity and Algebraic Compactness for Modules. Pacific J. Math., 1969, 28:699-719.
[24] Yang G, Liu Z K and Liang L. Ding Projective and Ding Injective Modules. Algebra Colloq, 2013,20:601-612.
[25] Zareh-Khoshchehreh F, Asgharzadeh M and Divaani-Aazar K. Gorenstein Homology, Relative Pure Homology and Virtually Gorenstein Rings. J. Pure Appl. Algebra, 2014, 218:2356-2366.
[2] Auslander M and Bridger M. Stable Module Theory. Memoirs. Amer. Math. Soc., Vol. 94,American Mathematical Society, Providence, RI, 1969.
[3] Bennis D. A note on Gorenstein flat dimension. Algebra Colloq., 2011, 18:155-161.
[4] Bennis D and Mahdou N. A Generalization of Strongly Gorenstein Projective Modules. J. Algebra Appl., 2009, 8:219-227.
[5] Bennis D and Mahdou N. Global Gorenstein Dimensions. Proc. Amer. Math. Soc., 2010, 138:461-465.
[6] Bennis D and Mahdou N. Strongly Gorenstein Projective, Injective, and Flat Modules. J. Pure Appl. Algebra, 2007, 210:437-445.
[7] Christensen L W. Gorenstein Dimensions. Lecture Notes in Math., Vol. 1747, Springer, Berlin,2000.
[8] Ding N Q and Chen J L. Coherent Rings with Finite Self-FP-injective Dimension. Comm. Algebra,1996, 24(9):2963-2980.
[9] Ding N Q and Chen J L. The Flat Dimensions of Injective Modules. Manuscripta Math., 1993, 78:165-177.
[10] Ding N Q, Li Y L and Mao L X. Strongly Gorenstein Flat Modules. J. Aust. Math. Soc., 2009, 86:323-338.
[11] Ding N Q and Mao L X. Relative FP-projective Modules. Comm. Algebra, 2005, 33:1587-1602.
[12] Enochs E E and Jenda O M G. Gorenstein Injective and Projective Modules. Math. Z., 1995, 220:611-633.
[13] Enochs E E and Jenda O M G. Relative Homological Algebra. Walter de Gruyter, Berlin, New York, 2000.
[14] Enochs E E, Jenda O M G and Torrecillas B. Gorenstein Flat Modules. Nanjing Daxue Xuebao Shuxue Bannian Kan,1993, 10:1-9.
[15] Fieldhouse D J. Character Modules, Dimension and Purity. Glasgow Math. J., 1972, 13:144-146.
[16] Gillespie J. Model Structures on Modules over Ding-Chen Rings. Homology, Homotopy Appl., 2010,12:61-73.
[17] Gillespie J. On Ding Injective, Ding Projective, and Ding Flat Modules and Complexes.ArXiv:1512.05999v1.
[18] G?bel R and Trlifaj J. Approximations and Endomorphism Algebras of Modules. Walter de Gruyter,Berlin, New York, 2006.
[19] Holm H. Gorenstein Homological Dimensions. J. Pure. Appl. Algebra, 2004, 189:167-193. Proc.Amer. Math. Soc. 1967, 18:155-158.
[20] Mahdou N and Tamekkante M. Strongly Gorenstein Flat Modules and Dimensions. Chin. Ann.Math., 2011, 32B(4):533-548.
[21] Stenstr?m B. Coherent Rings and FP-injective Modules. J. London Math. Soc., 1970, 2:323-329.
[22] Wang J, Li Y X and Hu J S. Noncommutative G-semihereditary Rings. J. Algebra Appl., 2018, 16:1850014, https://doi.org/10.1142/S0219498818500147.
[23] Warfield R R. Purity and Algebraic Compactness for Modules. Pacific J. Math., 1969, 28:699-719.
[24] Yang G, Liu Z K and Liang L. Ding Projective and Ding Injective Modules. Algebra Colloq, 2013,20:601-612.
[25] Zareh-Khoshchehreh F, Asgharzadeh M and Divaani-Aazar K. Gorenstein Homology, Relative Pure Homology and Virtually Gorenstein Rings. J. Pure Appl. Algebra, 2014, 218:2356-2366.