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C-Gorenstein projective, injective, flat modules and trivial ring extensions

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Veröffentlicht/Copyright: 17. Juni 2025
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Abstract

Let C be a semidualizing module over a commutative ring R and let R C be the trivial extension of R by C. We prove that (1) ( M , f ) is a Gorenstein projective R C -module if and only if M is a C-Gorenstein projective R-module; (2) [ M , g ] is a Gorenstein injective R C -module if and only if M is a C-Gorenstein injective R-module; (3) if R is a coherent ring, then ( M , f ) is a Gorenstein flat R C -module if and only if M is a C-Gorenstein flat R-module. Some applications of these results are given.

MSC 2020: 16D40; 16D50; 16E30

Award Identifier / Grant number: 12171230

Award Identifier / Grant number: 12271249

Funding statement: This research was supported by the National Natural Science Foundation of China (12171230, 12271249).

Acknowledgements

The author wants to express his gratitude to the referee for the very helpful comments and suggestions.

References

[1] M. Auslander and M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. 94, American Mathematical Society, Providence, 1969. 10.1090/memo/0094Suche in Google Scholar

[2] L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883. 10.1090/S0002-9947-01-02627-7Suche in Google Scholar

[3] N. Ding and J. Chen, Coherent rings with finite self-FP-injective dimension, Comm. Algebra 24 (1996), no. 9, 2963–2980. 10.1080/00927879608825724Suche in Google Scholar

[4] E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. 10.1007/BF02760849Suche in Google Scholar

[5] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633. 10.1007/BF02572634Suche in Google Scholar

[6] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Exp. Math. 30, Walter de Gruyter, Berlin, 2000. 10.1515/9783110803662Suche in Google Scholar

[7] 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. Suche in Google Scholar

[8] R. M. Fossum, P. A. Griffith and I. Reiten, Trivial Extensions of Abelian Categories, Lecture Notes in Math. 456, Springer, Berlin, 1975. 10.1007/BFb0065404Suche in Google Scholar

[9] H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284. 10.7146/math.scand.a-11434Suche in Google Scholar

[10] J. Gillespie, Model structures on modules over Ding–Chen rings, Homology Homotopy Appl. 12 (2010), no. 1, 61–73. 10.4310/HHA.2010.v12.n1.a6Suche in Google Scholar

[11] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, De Gruyter Exp. Math. 41, Walter de Gruyter, Berlin, 2006. 10.1515/9783110199727Suche in Google Scholar

[12] E. S. Golod, G-dimension and generalized perfect ideals (in Russian), Trudy Mat. Inst. Steklov. 165 (1984), 62–66. Suche in Google Scholar

[13] R. Hartshorne, Local Cohomology, Lecture Notes in Math. 41, Springer, Berlin, 1967. 10.1007/BFb0073971Suche in Google Scholar

[14] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1–3, 167–193. 10.1016/j.jpaa.2003.11.007Suche in Google Scholar

[15] H. Holm and P. Jørgensen, Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), no. 2, 423–445. 10.1016/j.jpaa.2005.07.010Suche in Google Scholar

[16] H. Holm and P. Jørgensen, Covers, precovers, and purity, Illinois J. Math. 52 (2008), no. 2, 691–703. 10.1215/ijm/1248355359Suche in Google Scholar

[17] H. Holm and D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ. 47 (2007), no. 4, 781–808. 10.1215/kjm/1250692289Suche in Google Scholar

[18] Z. Liu, Z. Huang and A. Xu, Gorenstein projective dimension relative to a semidualizing bimodule, Comm. Algebra 41 (2013), no. 1, 1–18. 10.1080/00927872.2011.602782Suche in Google Scholar

[19] L. Mao, Homological properties of trivial ring extensions, J. Algebra Appl. 22 (2023), no. 12, Article ID 2350265. 10.1142/S0219498823502651Suche in Google Scholar

[20] L. Mao, Pure projective, pure injective and FP-injective modules over trivial ring extensions, Internat. J. Algebra Comput. 33 (2023), no. 4, 699–716. 10.1142/S0218196723500315Suche in Google Scholar

[21] L. Mao, Silting and cosilting modules over trivial ring extensions, Comm. Algebra 51 (2023), no. 4, 1532–1550. 10.1080/00927872.2022.2137522Suche in Google Scholar

[22] L. Mao, A class of special formal triangular matrix rings, Bull. Malays. Math. Sci. Soc. 47 (2024), no. 4, Paper No. 129. 10.1007/s40840-024-01717-0Suche in Google Scholar

[23] J. Rada and M. Saorin, Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26 (1998), no. 3, 899–912. 10.1080/00927879808826172Suche in Google Scholar

[24] I. Reiten, Trivial extensions and Gorenstein rings, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1971. Suche in Google Scholar

[25] J. Šaroch and J. Šťovíček, Singular compactness and definability for Σ-cotorsion and Gorenstein modules, Selecta Math. (N. S.) 26 (2020), no. 2, Paper No. 23. 10.1007/s00029-020-0543-2Suche in Google Scholar

[26] B. Stenström, Coherent rings and F P -injective modules, J. Lond. Math. Soc. (2) 2 (1970), 323–329. 10.1112/jlms/s2-2.2.323Suche in Google Scholar

[27] R. Takahashi and D. White, Homological aspects of semidualizing modules, Math. Scand. 106 (2010), no. 1, 5–22. 10.7146/math.scand.a-15121Suche in Google Scholar

[28] W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Math. Stud. 14, North-Holland, Amsterdam, 1974. Suche in Google Scholar

[29] D. White, Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra 2 (2010), no. 1, 111–137. 10.1216/JCA-2010-2-1-111Suche in Google Scholar

[30] X. G. Yan and X. S. Zhu, Characterizations of some rings with 𝒞 -projective, 𝒞 -(FP)-injective and 𝒞 -flat modules, Czechoslovak Math. J. 61(136) (2011), no. 3, 641–652. 10.1007/s10587-011-0036-8Suche in Google Scholar

[31] X. Y. Yang and Z. K. Liu, C-Gorenstein projective, injective and flat modules, Czechoslovak Math. J. 60(135) (2010), no. 4, 1109–1129. 10.1007/s10587-010-0077-4Suche in Google Scholar

[32] D. Zhang and B. Ouyang, Semidualizing modules and related modules, J. Algebra Appl. 10 (2011), no. 6, 1261–1282. 10.1142/S0219498811005695Suche in Google Scholar

Received: 2024-11-10
Revised: 2025-02-04
Accepted: 2025-03-12
Published Online: 2025-06-17
Published in Print: 2025-02-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 5.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/gmj-2025-2045/pdf
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