• Yan, Hangyu
  • Received : 2009.04.10
  • Accepted : 2009.08.13
  • Published : 2010.09.30


In this paper, strongly cotorsion (torsion-free) modules are studied and strongly cotorsion (torsion-free) dimension is introduced. It is shown that every module has a special $\mathcal{SC}_n$-preenvelope and an ST $\mathcal{F}_n$-cover for any $n\;{\in}\;\mathbb{N}$ based on some results of cotorsion pairs from [9]. Some characterizations of strongly cotorsion (torsion-free) dimension of a module are given.


strongly cotorsion module;strongly torsion-free module;cotorsion pair;strongly cotorsion dimension;strongly torsion-free dimension


  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York-Heidelberg, 1974.
  2. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488.
  3. P. C. Eklof, S. Shelah, and J. Trlifaj, On the cogeneration of cotorsion pairs, J. Algebra 277 (2004), no. 2, 572-578.
  4. P. C. Eklof and J. Trlifaj, Covers induced by Ext, J. Algebra 231 (2000), no. 2, 640-651.
  5. P. C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. Lond. Math. Soc. 33 (2001), no. 1, 41-51.
  6. E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209.
  7. E. E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179-184.
  8. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter & Co., Berlin, 2000.
  9. R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, Walter de Gruyter GmbH & Co. KG, Berlin, 2006.
  10. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  11. L. Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), pp. 11-32, Academic Press, London-New York, 1979.
  12. R. Sazeedeh, Strongly torsion-free modules and local cohomology over Cohen-Macaulay rings, Comm. Algebra 33 (2005), no. 4, 1127-1135.
  13. J. Trlifaj, Infinite dimensional tilting modules and cotorsion pairs, Handbook of tilting theory, 279-321, London Math. Soc. Lecture Note Ser., 332, Cambridge Univ. Press, Cambridge, 2007.
  14. C. A. Weibel, An Introduction to Homological Algebra, Cambridge, Cambridge University Press, 1994.
  15. J. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag, Berlin, 1996.

Cited by

  1. Totally acyclic complexes vol.470, 2017,
  2. Gorenstein injective envelopes and covers over two sided noetherian rings vol.45, pp.5, 2017,