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Strongly \(\mathcal W\)-Gorenstein modules. (English) Zbl 1281.16019
Summary: Let \(\mathcal W\) be a self-orthogonal class of left \(R\)-modules. We introduce a class of modules, which is called strongly \(\mathcal W\)-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly \(\mathcal W\)-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly \(\mathcal W\)-Gorenstein module can be inherited by its submodules and quotient modules. As applications, many known results are generalized.

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16E05 Syzygies, resolutions, complexes in associative algebras
18G20 Homological dimension (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
Full Text: DOI
[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. 2. ed., Graduate Texts in Mathematics 13. Springer, New York, 1992. · Zbl 0765.16001
[2] M. Auslander, M. Bridger: Stable module theory. Mem. Am. Math. Soc. 94 (1969). · Zbl 0204.36402
[3] D. Bennis, N. Mahdou: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210 (2007), 437–445. · Zbl 1118.13014
[4] E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611–633. · Zbl 0845.16005
[5] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. Vol. 2. 2nd revised ed., de Gruyter Expositions in Mathematics 54. Walter de Gruyter, Berlin, 2000.
[6] E. E. Enochs, O. M. G. Jenda: On D-Gorenstein modules. Interactions between ring theory and representations of algebras. Proceedings of the conference, Murcia. Marcel Dekker, New York, 2000, pp. 159–168. · Zbl 0989.13018
[7] E. E. Enochs, O. M. G. Jenda: {\(\Omega\)}-Gorenstein projective and flat covers and {\(\Omega\)}-Gorenstein injective envelopes. Commun. Algebra 32 (2004), 1453–1470. · Zbl 1092.13031
[8] E. E. Enochs, O. M. G. Jenda, J. A. López-Ramos: Covers and envelopes by V-Gorenstein modules. Commun. Algebra 33 (2005), 4705–4717. · Zbl 1087.16002
[9] Y. Geng, N. Ding: W-Gorenstein modules. J. Algebra 325 (2011), 132–146. · Zbl 1216.18015
[10] S. Sather-Wagstaff, T. Sharif, D. White: Stability of Gorenstein categories. J. Lond. Math. Soc., II. Ser. 77 (2008), 481–502. · Zbl 1140.18010
[11] J. Wei: {\(\omega\)}-Gorenstein modules. Commun. Algebra 36 (2008), 1817–1829. · Zbl 1153.16009
[12] X. Yang, Z. Liu: Strongly Gorenstein projective, injective and flat modules. J. Algebra 320 (2008), 2659–2674. · Zbl 1173.16006
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