Covering classes, strongly flat modules, and completions


We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class \( \mathcal {SF}\) of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether \( \mathcal {SF}\) a covering class implies \( \mathcal {SF}\) closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.

This is a preview of subscription content, log in to check access.


  1. 1.

    Amini, B., Amini, A., Facchini, A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Anderson, D.W., Fuller, K.R.: Rings and categories of modules, 2nd edn. Springer, New York (1992)

    Google Scholar 

  3. 3.

    Angeleri Hügel, L., Sánchez, J.: Tilting modules arising from ring epimorphisms. Algebr. Represent. Theor. 14, 217–246 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bazzoni, S., Salce, L.: Strongly flat covers. J. Lond. Math. Soc. 66, 276–294 (2002)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bazzoni, S., Positselski, L.: \(S\)-almost perfect commutative rings. J. Algebra 532, 323–356 (2019)

  6. 6.

    Bazzoni, S., Positselski, L.: Contramodules over pro-perfect topological rings, the covering property in categorical tilting theory, and homological ring epimorphisms, available in arXiv:1807.10671

  7. 7.

    Bessenrodt, C., Brungs, H. H., Törner, G.: Right chain rings, Part 1, Schriftenreihe des Fachbereichs Math. 181 (Universität Duisburg, 1990)

  8. 8.

    Brungs, H.H., Dubrovin, N.I.: A classification and examples of rank one chain domains. Trans. Am. Math. Soc. 355, 2733–2753 (2003)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cohn, P.: Free ideal rings and localizations in general rings. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  10. 10.

    Dung, N.V., Facchini, A.: Direct summands of serial modules. J. Pure Appl. Algebra 133, 93–106 (1998)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dung, N.V., Facchini, A.: Weak Krull–Schmidt for infinite direct sums of uniserial modules. J. Algebra 193, 102–121 (1997)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Facchini, A.: Krull–Schmidt fails for serial modules. Trans. Am. Math. Soc. 348, 4561–4576 (1996)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Facchini, A., Nazemian, Z.: Equivalence of some homological conditions for ring epimorphism. J. Pure Appl. Algebra 223, 1440–1455 (2019)

  14. 14.

    Facchini, A., Salce, L.: Uniserial modules: sums and isomorphisms of subquotients. Comm. Algebra 18(2), 499–517 (1990)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fuchs, L., Salce, L.: Almost perfect commutative rings. J. Pure Appl. Algebra 222, 4223–4238 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Goodearl, K.R.: Ring theory; nonsingular rings and modules. Dekker, New York (1976)

    Google Scholar 

  17. 17.

    Goodearl, K.R., Warfield, R.B.: An introduction to noncommutative noetherian rings, 2nd edn. Cambridge Univ. Press, Cambridge (2004)

    Google Scholar 

  18. 18.

    Göbel, R., Trlifaj, J.: Approximations and endomorphism algebras of modules. Walter de Gruyter, Berlin (2006)

    Google Scholar 

  19. 19.

    Lam, T.Y.: Lectures on modules and rings. Springer, New York (1999)

    Google Scholar 

  20. 20.

    Matlis, E.: 1 -dimensional Cohen–Macaulay rings. Springer, Berlin, New York (1973)

    Google Scholar 

  21. 21.

    Nicholson, W.K., Yousif, M.F.: Quasi-Frobenius rings. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  22. 22.

    Positselski, L.: Flat ring epimorphisms of countable type, available in arXiv:1808.00937

  23. 23.

    Příhoda, P.: \({\rm Add}(U)\) of a uniserial module. Comment. Math. Univ. Carolin. 47, 391–398 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Puninski, G.: Some model theory over a nearly simple uniserial domain and decompositions of serial modules. J. Pure Appl. Algebra 163, 319–337 (2001)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Stenström, B.: Rings of quotients. Springer, New York (1975)

    Google Scholar 

  26. 26.

    Schofield, A.H.: Representations of rings over skew fields. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  27. 27.

    Warfield, R.B.: Purity and algebraic compactness for modules. Pacific J. Math. 28, 699–719 (1969)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wisbauer, R.: Foundations of module and ring theory. Gordon and Breach, Philadelphia (1991)

    Google Scholar 

  29. 29.

    Xu, J.: Flat covers of modules. Lecture notes in mathematics, vol. 1634. Springer, New York (1996)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Alberto Facchini.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was partially supported by Dipartimento di Matematica “Tullio Levi-Civita” of Università di Padova (Project BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and Research program DOR1714214 “Anelli e categorie di moduli”). The second author was supported by a Grant from IPM.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Facchini, A., Nazemian, Z. Covering classes, strongly flat modules, and completions. Math. Z. 296, 239–259 (2020).

Download citation


  • Covering class
  • Strongly flat module
  • Completion
  • Cotorsion module
  • R-topology

Mathematics Subject Classification

  • Primary 16E30
  • 16W80
  • Secondary 18G15