Acknowledgement
The authors would like to express their gratitude to an anonymous reviewer of this paper for his insightful comments and suggestions which have greatly improved the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A3B03033342).
References
- F. A. A. Almahdi, M. Tamekkante, and R. A. K. Assaad, On the right orthogonal complement of the class of w-flat modules, J. Ramanujan Math. Soc. 33 (2018), no. 2, 159-175.
- D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2010), no. 2, 461-465. https://doi.org/10.1090/S0002-9939-09-10099-0
- H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
- R. R. Colby, Rings which have flat injective modules, J. Algebra 35 (1975), 239-252. https://doi.org/10.1016/0021-8693(75)90049-6
- N. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math. 78 (1993), no. 2, 165-177. https://doi.org/10.1007/BF02599307
- N. Ding and J. L. Chen, On copure flat modules and flat resolvents, Comm. Algebra 24 (1996), no. 3, 1071-1081. https://doi.org/10.1080/00927879608825623
- E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolvents and dimensions, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 203-211.
- E. E. Enochs and O. M. G. Jenda, On Gorenstein injective modules, Comm. Algebra 21 (1993), no. 10, 3489-3501. https://doi.org/10.1080/00927879308824744
- 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, O. M. G. Jenda, and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), no. 1, 1-9.
- X. Fu and N. Ding, On strongly copure flat modules and copure flat dimensions, Comm. Algebra 38 (2010), no. 12, 4531-4544. https://doi.org/10.1080/00927870903428262
- S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/BFb0084570
- S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math. 22 (1977), no. 4, 325-341. https://doi.org/10.1007/BF01168220
- 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
- K. Hu, F. G. Wang, and L. Xu, A note on Gorenstein Prufer domains, Bull. Korean Math. Soc. 53 (2016), no. 5, 1447-1455. https://doi.org/10.4134/BKMS.b150760
- H. Kim and F. G. Wang, On LCM-stable modules, J. Algebra Appl. 13 (2014), no. 4, 1350133, 18 pp. https://doi.org/10.1142/S0219498813501338
- N. Mahdou and M. Tamekkante, On (weak) Gorenstein global dimensions, Acta Math. Univ. Comenian. (N.S.) 82 (2013), no. 2, 285-296.
- A. Mimouni, Integral domains in which each ideal is a W-ideal, Comm. Algebra 33 (2005), no. 5, 1345-1355. https://doi.org/10.1081/AGB-200058369
- F. G. Wang, On w-projective modules and w-flat modules, Algebra Colloq. 4 (1997), no. 1, 111-120.
- F. G. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ., 33 (2010), 1-9. https://doi.org/10.3969/j.issn.1001-8395.2010.01.001
- F. G. Wang and H. Kim, w-injective modules and w-semi-hereditary rings, J. Korean Math. Soc. 51 (2014), no. 3, 509-525. https://doi.org/10.4134/JKMS.2014.51.3.509
- F. G. Wang and H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra 219 (2015), no. 6, 2099-2123. https://doi.org/10.1016/j.jpaa.2014.07.025
- F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016. https://doi.org/10.1007/978-981-10-3337-7
- F. G. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), no. 4, 1285-1306. https://doi.org/10.1080/00927879708825920
- F. G. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean Math. Soc. 52 (2015), no. 4, 1327-1338. https://doi.org/10.4134/BKMS.2015.52.4.1327
- F. G. Wang and L. Qiao, A homological characterization of Krull domains II, Comm. Algebra 47 (2019), no. 5, 1917-1929. https://doi.org/10.1080/00927872.2018.1524007
- L. Xie, F. G. Wang, and Y. Tian, On w-linked overrings, J. Math. Res. Exposition 31 (2011), no. 2, 337-346.
- T. Xiong, F. G. Wang, G. L. Xia, and X. W. Sun, Change theorem of rings on the copure flat dimensions, J. Nat. Sci. Heilongjiang Univ. 33 (2016), 435-437.
- J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, 1634, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/BFb0094173
- H. Yin, F. G.Wang, X. Zhu, and Y. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 207-222. https://doi.org/10.4134/JKMS.2011.48.1.207
- S. Q. Zhao, F. G. Wang, and H. L. Chen, Flat modules over a commutative ring are w-modules, J. Sichuan Normal Univ. 35 (2012), 364-366. https://doi.org/10.3969/j.issn.1001-8395.2012.03.016