Advertisement

Covering classes, strongly flat modules, and completions

  • Alberto FacchiniEmail author
  • Zahra Nazemian
Article
  • 12 Downloads

Abstract

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.

Keywords

Covering class Strongly flat module Completion Cotorsion module R-topology 

Mathematics Subject Classification

Primary 16E30 16W80 Secondary 18G15 

Notes

References

  1. 1.
    Amini, B., Amini, A., Facchini, A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Anderson, D.W., Fuller, K.R.: Rings and categories of modules, 2nd edn. Springer, New York (1992)CrossRefGoogle Scholar
  3. 3.
    Angeleri Hügel, L., Sánchez, J.: Tilting modules arising from ring epimorphisms. Algebr. Represent. Theor. 14, 217–246 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bazzoni, S., Salce, L.: Strongly flat covers. J. Lond. Math. Soc. 66, 276–294 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bazzoni, S., Positselski, L.: \(S\)-almost perfect commutative rings. J. Algebra 532, 323–356 (2019)Google Scholar
  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)Google Scholar
  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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cohn, P.: Free ideal rings and localizations in general rings. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  10. 10.
    Dung, N.V., Facchini, A.: Direct summands of serial modules. J. Pure Appl. Algebra 133, 93–106 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dung, N.V., Facchini, A.: Weak Krull–Schmidt for infinite direct sums of uniserial modules. J. Algebra 193, 102–121 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Facchini, A.: Krull–Schmidt fails for serial modules. Trans. Am. Math. Soc. 348, 4561–4576 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Facchini, A., Nazemian, Z.: Equivalence of some homological conditions for ring epimorphism. J. Pure Appl. Algebra 223, 1440–1455 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Facchini, A., Salce, L.: Uniserial modules: sums and isomorphisms of subquotients. Comm. Algebra 18(2), 499–517 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fuchs, L., Salce, L.: Almost perfect commutative rings. J. Pure Appl. Algebra 222, 4223–4238 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goodearl, K.R.: Ring theory; nonsingular rings and modules. Dekker, New York (1976)zbMATHGoogle Scholar
  17. 17.
    Goodearl, K.R., Warfield, R.B.: An introduction to noncommutative noetherian rings, 2nd edn. Cambridge Univ. Press, Cambridge (2004)CrossRefGoogle Scholar
  18. 18.
    Göbel, R., Trlifaj, J.: Approximations and endomorphism algebras of modules. Walter de Gruyter, Berlin (2006)CrossRefGoogle Scholar
  19. 19.
    Lam, T.Y.: Lectures on modules and rings. Springer, New York (1999)CrossRefGoogle Scholar
  20. 20.
    Matlis, E.: 1 -dimensional Cohen–Macaulay rings. Springer, Berlin, New York (1973)CrossRefGoogle Scholar
  21. 21.
    Nicholson, W.K., Yousif, M.F.: Quasi-Frobenius rings. Cambridge University Press, Cambridge (2003)CrossRefGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Stenström, B.: Rings of quotients. Springer, New York (1975)CrossRefGoogle Scholar
  26. 26.
    Schofield, A.H.: Representations of rings over skew fields. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  27. 27.
    Warfield, R.B.: Purity and algebraic compactness for modules. Pacific J. Math. 28, 699–719 (1969)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wisbauer, R.: Foundations of module and ring theory. Gordon and Breach, Philadelphia (1991)zbMATHGoogle Scholar
  29. 29.
    Xu, J.: Flat covers of modules. Lecture notes in mathematics, vol. 1634. Springer, New York (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations