Czechoslovak Mathematical Journal

, Volume 62, Issue 3, pp 663–672 | Cite as

k-torsionless modules with finite Gorenstein dimension

  • Maryam Salimi
  • Elham Tavasoli
  • Siamak Yassemi


Let R be a commutative Noetherian ring. It is shown that the finitely generated R-module M with finite Gorenstein dimension is reflexive if and only if M p is reflexive for p ∈ Spec(R) with depth(R p) ⩽ 1, and \(G - {\dim _{{R_p}}}\) (M p) ⩽ depth(R p) − 2 for p ∈ Spec(R) with depth(R p) ⩾ 2. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n ⩾ 2 we give a characterization of n-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every R-module has a k-torsionless cover provided R is a k-Gorenstein ring.


torsionless module reflexive module Gorenstein dimension 

MSC 2010



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  1. [1]
    M. Auslander, M. Bridger: Stable module theory. Mem. Am. Math. Soc. 94 (1969).Google Scholar
  2. [2]
    L. L. Avramov, S.B. Iyengar, J. Lipman: Reflexivity and rigidity for complexes, I: Commutative rings. Algebra Number Theory 4 (2010), 47–86.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    R. Belshoff: Remarks on reflexive modules, covers, and envelopes. Beitr. Algebra Geom. 50 (2009), 353–362.MathSciNetMATHGoogle Scholar
  4. [4]
    W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge University Press, Cambridge, 1993.MATHGoogle Scholar
  5. [5]
    L.W. Christensen, H. Holm: Ascent properties of Auslander categories. Can. J. Math. 61 (2009), 76–108.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    L.W. Christensen, H.B. Foxby, H. Holm: Beyond Totally Reflexive Modules and Back: A Survey on Gorenstein Dimensions. Marco Fontana, Commutative algebra. Noetherian and non-Noetherian perspectives. New York, 2011, pp. 101–143.Google Scholar
  7. [7]
    E. Enochs, O.M.G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics. 30. Berlin: Walter de Gruyter. xi, 2000.CrossRefGoogle Scholar
  8. [8]
    C. Huneke, R. Wiegand: Tensor products of modules and the rigidity of Tor. Math. Ann. 299 (1994), 449–476.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    C. Huneke, R. Wiegand: Correction to “Tensor products of modules and the rigidity of Tor”. [Math. Ann. 299, 449–476 (1994)]. Math. Ann. 338 (2007), 291–293.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    V. Maşek: Gorenstein dimension and torsion of modules over commutative Noetherian rings. Commun. Algebra 28 (2000), 5783–5811.MATHCrossRefGoogle Scholar
  11. [11]
    P. Samuel: Anneaux gradués factoriels et modules réflexifs. Bull. Soc. Math. Fr. 92 (1964), 237–249. (In French.)MathSciNetMATHGoogle Scholar
  12. [12]
    J.-P. Serre: Classes des corps cyclotomiques. Semin. Bourbaki. vol. 11, 1958/59, pp. 11.Google Scholar
  13. [13]
    W. Vasconcelos: Reflexive modules over Gorenstein rings. Proc. Am. Math. Soc. 19 (1968), 1349–1355.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of MathematicsUniversity of Tehran, and School of mathematics, Institute for research in fundamental sciences (IPM)TehranIran

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