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Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 55–73 | Cite as

n-Strongly Gorenstein Graded Modules

  • Zenghui GaoEmail author
  • Jie Peng
Article
  • 35 Downloads

Abstract

Let R be a graded ring and n ⩾ 1 an integer. We introduce and study n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that n-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be m-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever n > m. Many properties of the n-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate the relations between the graded and the ungraded n-strongly Gorenstein injective (or flat) modules. In addition, the connections between the n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules are considered.

Keywords

n-strongly Gorenstein gr-injective module n-strongly Gorenstein gr-flat module n-strongly Gorenstein gr-projective module 

MSC 2010

16W50 18G25 16E05 

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References

  1. [1]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-injective and gr-projective modules. Commun. Algebra 26 (1998), 225–240.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-flat modules. Commun. Algebra 26 (1998), 3195–3209.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: Covers and envelopes over gr-Gorenstein rings. J. Algebra 215 (1999), 437–459.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: FP-gr-injective modules and gr-FC-rings. Algebra and Number Theory. Proc. Conf., Fez, Morocco (M. Boulagouaz, ed.). Lecture Notes in Pure and Appl. Math. 208, Marcel Dekker, New York, 2000, pp. 1–11.zbMATHGoogle Scholar
  5. [5]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein gr-injective modules over graded isolated singularities. Commun. Algebra 28 (2000), 3197–3207.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. J. Asensio, J. A. López Ramos, B. Torrecillas: Gorenstein modules over Zariski fil-tered rings. Commun. Algebra 31 (2003), 4371–4385.CrossRefzbMATHGoogle Scholar
  7. [7]
    M. Auslander, M. Bridger: Stable Module Theory. Memoirs of the American Mathemat-ical Society 94, American Mathematical Society, Providence, 1969.CrossRefzbMATHGoogle Scholar
  8. [8]
    D. Bennis, N. Mahdou: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210 (2007), 437–445.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Bennis, N. Mahdou: A generalization of strongly Gorenstein projective modules. J. Algebra Appl. 8 (2009), 219–227.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    L. W. Christensen: Gorenstein Dimensions. Lecture Notes in Mathematics 1747. Springer, Berlin, 2000.CrossRefzbMATHGoogle Scholar
  11. [11]
    N. Q. Ding, J. L. Chen: The flat dimensions of injective modules. Manuscr. Math. 78 (1993), 165–177.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Q. Ding, J. L. Chen: Coherent rings with finite self-FP-injective dimension. Commun. Algebra 24 (1996), 2963–2980.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611–633.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. de De Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin, 2000.CrossRefzbMATHGoogle Scholar
  15. [15]
    E. E. Enochs, O. M. G. Jenda, B. Torrecillas: Gorenstein flat modules. J. Nanjing Univ., Math. Biq. 10 (1993), 1–9.MathSciNetzbMATHGoogle Scholar
  16. [16]
    E. E. Enochs, J. A. López Ramos: Gorenstein Flat Modules. Nova Science Publishers, Huntington, 2001.zbMATHGoogle Scholar
  17. [17]
    J. R. García Rozas, J. A. López-Ramos, B. Torrecillas: On the existence of flat covers in R-gr. Commun. Algebra 29 (2001), 3341–3349.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Hermann, S. Ikeda, U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study. Springer, Berlin, 1988.CrossRefzbMATHGoogle Scholar
  19. [19]
    H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167–193.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    L. X. Mao: Strongly Gorenstein graded modules. Front. Math. China 12 (2017), 157–176.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C. Năstăsescu: Some constructions over graded rings: Applications. J. Algebra 120 (1989), 119–138.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    C. Năstăsescu, F. Van Oystaeyen: Graded Ring Theory. North-Holland Mathematical Library 28, North-Holland Publishing Company, Amsterdam, 1982.zbMATHGoogle Scholar
  23. [23]
    C. Năstăsescu, F. Van Oystaeyen: Methods of Graded Rings. Lecture Notes in Mathe-matics 1836, Springer, Berlin, 2004.zbMATHGoogle Scholar
  24. [24]
    B. Stenström: Rings of Quotients. Die Grundlehren der mathematischen Wissenschaf-ten 217. Springer, Berlin, 1975. (In German.)CrossRefzbMATHGoogle Scholar
  25. [25]
    X. Yang, Z. Liu: Strongly Gorenstein projective, injective and flat modules. J. Algebra 320 (2008), 2659–2674.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    X. Yang, Z. Liu: FP-gr-injective modules. Math. J. Okayama Univ. 53 (2011), 83–100.MathSciNetzbMATHGoogle Scholar
  27. [27]
    G. Q. Zhao, Z. Y. Huang: n-strongly Gorenstein projective, injective and flat modules. Commun. Algebra 39 (2011), 3044–3062.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.College of Applied MathematicsChengdu University of Information TechnologyChengduP.R. China

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