Acknowledgement
Supported by : NSFC, Jiangsu University of Technology of China
References
- F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
- M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.
- R. G. Belshoff, Matlis reflexive modules, Comm. Algebra 19 (1991), no. 4, 1099-1118. https://doi.org/10.1080/00927879108824192
- R. G. Belshoff, Remarks on reflexive modules, covers, and envelopes, Beitrage Algebra Geom. 50 (2009), no. 2, 353-362.
- R. G. Belshoff, E. E. Enochs, and J. R. Garcia Rozas, Generalized Matlis duality, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1307-1312. https://doi.org/10.1090/S0002-9939-99-05130-8
- R. G. Belshoff and J. Z. Xu, Injective envelopes and flat covers of Matlis reflexive modules, J. Pure Appl. Algebra 79 (1992), no. 3, 205-215. https://doi.org/10.1016/0022-4049(92)90050-P
- W. Bruns and J. Herzog, Cohen-Macaulay Rings (revised edition), Advances in Math. 39, Cambridge Univ. Press, Cambridge, UK, 1996.
- L. W. Christensen, Gorenstein Dimension, Lecture Notes in Math. 1747, Springer-Verlag, Berlin, 2000.
- N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470. https://doi.org/10.1080/00927879608825646
- E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209. https://doi.org/10.1007/BF02760849
- E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Compact coGalois groups, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 233-244. https://doi.org/10.1017/S0305004199004156
- E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Covering morphisms, Comm. Algebra 28 (2000), no. 8, 3823-3835. https://doi.org/10.1080/00927870008827060
- E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Are covering (enveloping) morphisms minimal?, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2863-2868. https://doi.org/10.1090/S0002-9939-00-05339-9
- E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Finitely generated cotorsion modules, Proc. Edinb. Math. Soc. 44 (2001), no. 1, 143-152. https://doi.org/10.1017/S0013091599000590
- E. E. Enochs and Z. Y. Huang, Injective envelopes and (Gorenstein) flat covers, Algebr. Represent. Theory 15 (2012), no. 6, 1131-1145. https://doi.org/10.1007/s10468-011-9282-6
- E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611-633. https://doi.org/10.1007/BF02572634
- E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
- E. E. Enochs, O. M. G. Jenda, and J. Z. Xu, A generalization of Auslander's last theorem, Algebr. Represent. Theory 2 (1999), no. 3, 259-268. https://doi.org/10.1023/A:1009998109625
- M. Hashimoto and A. Shida, Some remarks on index and generalized Loewy length of a Gorenstein local ring, J. Algebra 187 (1997), no. 1, 150-162. https://doi.org/10.1006/jabr.1997.6770
- H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167-193. https://doi.org/10.1016/j.jpaa.2003.11.007
- Z. Y. Huang, Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), 142-169. https://doi.org/10.1016/j.jalgebra.2013.07.008
- Z. Y. Huang, Duality of preenvelopes and pure injective modules, Canad. Math. Bull. 57 (2014), no. 2, 318-325. https://doi.org/10.4153/CMB-2013-023-x
- D. A. Jorgensen and L. M. Sega, Independence of the total reflexivity conditions for modules, Algebr. Represent. Theory 9 (2006), no. 2, 217-226. https://doi.org/10.1007/s10468-005-0559-5
- W. V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349-1355. https://doi.org/10.1090/S0002-9939-1968-0237480-2
- W. M. Xue, Injective envelopes and flat covers of modules over a commutative ring, J. Pure Appl. Algebra 109 (1996), no. 2, 213-220. https://doi.org/10.1016/0022-4049(95)00085-2
- Y. Yoshino, Cohen-Macaulay approximations, In: Proc. symposium on representation theory of algebras (Izu, Japan, 1993) pp. 119-138 (in Japanese).