Abstract
By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).
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References
H. Bass, On the ubiquity of Gorenstein rings, Mathematische Zeitschrift 82 (1963), 8–28.
L. Bican, R. El Bashir and E. E. Enochs, All modules have flat covers, The Bulletin of the London Mathematical Society 33 (2001), 385–390.
J. E. Björk, The Auslander condition on Noetherian rings, in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, Paris, 1987/1988, Lecture Notes in Mathematics, Vol. 1404, Springer-Verlag, Berlin, 1989, pp.137–173.
H. Cartan and S. Eilenberg, Homological Algebra, Reprint of the 1956 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1999.
N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Mathematica 78 (1993), 165–177.
E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel Journal of Mathematics 39 (1981), 189–209.
E. E. Enochs, Flat covers and flat cotorsion modules, Proceedings of the American Mathematical Society 92 (1984), 179–184.
E. E. Enochs and Z. Y. Huang, Injective envelopes and (Gorenstein) flat covers, Algebras and Representation Theory 15 (2012), 1131–1145.
E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Mathematische Zeitschrift 220 (1995), 611–633.
E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, Vol. 30, Walter de Gruyter, Berlin, New York, 2000.
E. E. Enochs, O. M. G. Jenda and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao. Shuxue Bannian Kan. Nanjing University. Journal. Mathematical Biquarterly 10 (1993), 1–9.
D. J. Fieldhouse, Character modules, Commentarii Mathematici Helvetici 46 (1971), 274–276.
R. M. Fossum, P. A. Griffith and I. Reiten, Trivial Extensions of Abelian Categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin, 1975.
H. Holm, Gorenstein homological dimensions, Journal of Pure and Applied Algebra 189 (2004), 167–193.
Y. Iwanaga, On rings with finite self-injective dimension II, Tsukuba Journal of Mathematics 4 (1980), 107–113.
K. Khashyarmanesh and Sh. Salarian, On the rings whose injective hulls are flat, Proceedings of the American Mathematical Society 131 (2003), 2329–2335.
L. X. Mao and N. Q. Ding, Envelopes and covers by modules of finite FP-injective and flat dimensions, Communications in Algebra 35 (2007), 833–849.
J. Z. Xu, Minimal injective and flat resolutions over Gorenstein rings, Journal of Algebra 175 (1995), 451–477.
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Huang, Z. Gorenstein injective and strongly cotorsion modules. Isr. J. Math. 198, 215–228 (2013). https://doi.org/10.1007/s11856-013-0033-8
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DOI: https://doi.org/10.1007/s11856-013-0033-8