Abstract
In this paper, we explore conditions under which Gorenstein flat modules are Gorenstein projective. We prove that all countably presented strongly Gorenstein flat modules are Gorenstein projective over perfect rings. Moreover, we show that if the base ring R is \(\sum \)-pure injective as an R-module, then the class of Gorenstein flat modules coincides with the class of Gorenstein projective modules, and hence all modules have Gorenstein projective covers. And as a corollary, we give a characterization of coherent perfect rings by Gorenstein projective and Gorenstein flat modules.
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References
Angeleri Hügel, L., Herbera, D.: Mittag–Leffler conditions on modules. Indiana Univ. Math. J. 57(5), 2459–2517 (2008)
Auslander, M., Bridger, M.: Stable module theory. Mem. Am. Math. Soc. 94, 92–146 (1969)
Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective and flat modules. J. Pure. Appl. Algebra 210(2), 437–445 (2007)
Emmanouil, I., Talelli, O.: On the flat length of injective modules. J. Lond. Math. Soc. 84(2), 408–432 (2011)
Emmanouil, I.: On the finiteness of Gorenstein homological dimensions. J. Algebra 372, 376–396 (2012)
Enochs, E.E., Jenda, O.: Gorenstein injective and projective modules. Math. Z. 220(4), 611–633 (1995)
Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra, De Gruyter Exp. Math., vol. 30. Walter de Gruyter, Berlin (2000)
Enochs, E.E., Jenda, O., Torrecillas, B.: Gorenstein flat modules. J. Nanjing Univ. Math. Biquarterly 10(1), 1–9 (1993)
Estrada, S., Iacob, A., Odabasi, S.: Gorenstein flat and projective (pre) covers. Publ. Math. 91(1), 111–121 (2017)
Grothendieck, A.: EGA III. Publ. Math. Inst. Hautes Etudes Sci. 11, 17 (1961)
Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules, De Gruyter Exp. Math., vol. 41. Walter de Gruyter, Berlin (2006)
Yang, G., Liang, L.: All modules have Gorenstein flat precovers. Commun. Algebra 42(7), 3078–3085 (2014)
Yang, Y., Yan, X., Zhu, X.: Strict Mittag–Leffler conditions and Gorenstein modules. Algebr. Represent. Theory 19(6), 1451–1466 (2016)
Yang, Y., Zhu, X., Yan, X.: Strict Mittag–Leffler conditions on Gorenstein modules. Commun. Algebra 44(4), 1754–1766 (2016)
Zimmermann, W.: Modules with chain conditions for finite matrix subgroups. J. Algebra 190(1), 68–87 (1997)
Acknowledgements
This research was supported by NFSC (No. 11701408 and No. 11571165) and NSF of the Jiangsu Higher Education Institution (No. 16KJD1100 04). The authors would like to thank the referee for the very helpful comments and suggestions.
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Communicated by Mohammad-Taghi Dibaei.
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Yang, Y., Yan, X. Notes on Gorenstein Flat Modules. Bull. Iran. Math. Soc. 45, 337–344 (2019). https://doi.org/10.1007/s41980-018-0135-5
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DOI: https://doi.org/10.1007/s41980-018-0135-5