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Strict Mittag-Leffler Conditions and Gorenstein Modules

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Abstract

In this paper, firstly, we characterize some rings by strict Mittag-Leffler conditions. Then, we investigate when Gorenstein projective modules are Gorenstein flat by employing tilting modules and cotorsion pairs. Finally, we study the direct limits of Gorenstein projective modules.

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References

  1. Hügel, L.A., Herbera, D.: Mittag-Leffler conditions on modules. Indiana Math. J. 57, 2459–2517 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hügel, L.A., Bazzoni, S., Herbera, D.: A solution to the Baer splitting problem. Trans. Amer. Math. Soc. 360, 2409–2421 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hügel, L.A., Šaroch, J., Trlifaj, J.: Approximations and Mittag-Leffler conditions. preprint

  4. Auslander, M., Bridger, M.: Stable module theory. Mem. Amer. Math. Soc. 94 (1969)

  5. Azumaya, G.: Locally pure-projective modules. Contemp. Math. 124, 17–22 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Azumaya, G., Facchini, A.: Rings of pure global dimension zero and Mittag-Leffler modules. J. Pure Appl. Algebra 62, 109–122 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bazzoni, S., Herbera, D.: One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11, 43–61 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bazzoni, S., Šťovíček, J.: All tilting modules are of finite type. Proc. Amer. Math. Soc. 135, 3771–3781 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective and flat modules. J. Pure. Appl. Algebra 210, 437–445 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, N., Chen, J.: Coherent ring with finite self-FP-injective dimension. Comm. Algebra 24, 2963–2980 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, N., Li, Y., Mao, L.: Strongly Gorenstein flat modules. J. Aust. Math. Soc. 86, 323–338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Emmanouil, I.: On certain cohomological invariants of groups. Adv. Math. 225, 3446–3462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Emmanouil, I., Talelli, O.: On the flat length of injective modules. J. Lond. Math. Soc. 84, 408–432 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Emmanouil, I.: On the finiteness of Gorenstein homological dimensions. J. Algebra 372, 376–396 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Enochs, E.E., Jenda, O.: Relative Homological Algebra, de Gruyter Exp. Math., vol. 30, Walter de Gruyter and Co., Berlin, 2000

  16. Enochs, E.E., Jenda, O.: Gorenstein injective and projective modules. Math. Z 220(4), 611–633 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Enochs, E.E., Jenda, O., Torrecillas, B.: Gorenstein flat modules. Nan**g Daxue Xuebao Shuxue Bannian Kan 10(1), 1–9 (1993)

    MathSciNet  MATH  Google Scholar 

  18. Enochs, E.E., López-Ramos, J.A.: Kaplansky classes. Rend. Sem. Mat. Univ. Padova. 107, 67–79 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules, de Gruyter Exp. Math., vol. 41, W. de Gruyter, Berlin (2006)

  20. Grothendieck, A.: EGA, III. Publ. Math. Inst. Hautes Etudes Scient. 11 (1961)

  21. Herbera, D.: Definable classes and Mittag-Leffler conditions. Ring Theory and Its Appl. Contemporary Math. 609, 137–166 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Herbera, D., Trlifaj, J.: Almost free modules and Mittag-Leffler conditions. Adv. Math. 229, 3436–3467 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Holm, H., Gorenstein homological dimensions: J. Pure Appl. Algebra 189, 167–193 (2004)

    Google Scholar 

  24. Holm, H., Jorgensen, P.: Rings without a gorenstein analogue of the Govorov-Lazard theorem. Q. J. Math. 62(4), 977–988 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jensen, C.U.: On the vanishing of \(\underset {\leftarrow }{\lim }^{(i)}\). J. Algebra 15, 151–166 (1970)

    Article  MathSciNet  Google Scholar 

  26. Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Invent. Math. 13, 1–89 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stenström, B.: Coherent rings and FP-injective modules. J. London Math. Soc. 2, 323–329 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  28. Šaroch, J., Šťovíček, J.: The countable Telescope Conjecture for module categories. Adv. Math. 219, 1002–1036 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, J., Liang, L.: A Characterization of Gorenstein projective modules. preprint

  30. Xu, J.: Flat Covers of Modules, Lecture Notes in Mathematics 1634; Springer Verlag: Berlin-Heidelberg-NewYork (1996)

  31. Yang, G., Liu, Z.K.: Gorenstein flat covers over GF-closed rings. Comm. Algebra 40, 1632–1640 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, Y., Zhu, X., Yan, X.: Strict Mittag-Leffler conditions on Gorenstein modules, to appear in Comm. Algebra

  33. Zhang, P.: A brief introduction to Gorenstein projective modules, http://www.mathematik.uni-bielefeld.de/~sek/sem/abs/zhangpu4.pdf

  34. Zimmermann, W.: Modules with chain conditions for finite matrix subgroups. J. Algebra 190, 68–87 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nan**g University, Nan**g 210093, China

    Yanjiong Yang & **aosheng Zhu

  2. School of Mathematics and Information Technology, Nan**g **aozhuang University, Nan**g 211171, China

    **aoguang Yan

  3. Department of Mathematics, Taizhou University, Taizhou 225300, China

    Yanjiong Yang

Authors
  1. Yanjiong Yang
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  2. **aoguang Yan
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  3. **aosheng Zhu
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Correspondence to Yanjiong Yang.

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Presented by Steffen Koenig.

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Yang, Y., Yan, X. & Zhu, X. Strict Mittag-Leffler Conditions and Gorenstein Modules. Algebr Represent Theor 19, 1451–1466 (2016). https://doi.org/10.1007/s10468-016-9626-3

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  • DOI: https://doi.org/10.1007/s10468-016-9626-3

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