Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

RINGS WITH VARIATIONS OF FLAT COVERS

  • Received : 2019.03.12
  • Accepted : 2019.06.04
  • Published : 2019.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce flat e-covers of modules and define e-perfect rings as a generalization of perfect rings. We prove that a ring is right perfect if and only if it is semilocal and right e-perfect which generalizes a result due to N. Ding and J. Chen. Moreover, in the light of the fact that a ring R is right perfect if and only if flat covers of any R-module are projective covers, we study on the rings over which flat covers of modules are (generalized) locally projective covers, and obtain some characterizations of (semi) perfect, A-perfect and B-perfect rings.

Keywords

Acknowledgement

The authors would like to thank the referees for careful reading of the paper and valuable suggestions.

References

  1. A. Amini, B. Amini, M. Ershad and H. Sharif, On generalized perfect rings, Comm. Algebra 35 (2007), no. 3, 953-963. https://doi.org/10.1080/00927870601115880
  2. A. Amini, M. Ershad and H. Sharif, Rings over which flat covers of finitely generated modules are projective, Comm. Algebra 36 (2008), no. 8, 2862-2871. https://doi.org/10.1080/00927870802108049
  3. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992.
  4. P. Aydogdu, Rings over which every module has a flat ${\delta}$-cover, Turkish J. Math. 37 (2013), no. 1, 182-194.
  5. G. Azumaya, Some characterizations of regular modules, Publ. Mat. 34 (1990), 241-248. https://doi.org/10.5565/PUBLMAT_34290_02
  6. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. https://doi.org/10.1090/S0002-9947-1960-0157984-8
  7. L.Bican, R. El Bashir and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385-390. https://doi.org/10.1017/S0024609301008104
  8. E. Buyukasik, Rings over which flat covers of simple modules are projective, J. Algebra Appl. 11 (2012), no. 3, 1250046. https://doi.org/10.1142/S0219498811005737
  9. E. Buyukasik and C. Lomp, When ${\delta}$-semiperfect rings are semiperfect, Turkish J. Math. 34 (2010), no. 3, 317-324.
  10. E. Buyukasik and D. Pusat-Yilmaz, Modules whose maximal submodules are supplements. Hacet. J. Math. Stat. 39 (2010), no. 4, 477-487.
  11. Y. M. Demirci, On generalizations of semiperfect and perfect rings, Bull. Iran. Math. Soc. 42 (2016), no. 6, 1441-1450.
  12. N. Ding and J. Chen, On a characterization of perfect rings, Comm. Algebra 27 (1999), no. 2, 785-791. https://doi.org/10.1080/00927879908826461
  13. E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209. https://doi.org/10.1007/BF02760849
  14. P. A. Guil Asensio and I. Herzog, Indecomposable flat cotorsion modules, J. London Math. Soc. 76 (2007), no. 3, 797. https://doi.org/10.1112/jlms/jdm076
  15. F. Kasch, Modules and rings, London Mathematical Society Monographs 17, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982.
  16. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1999.
  17. C. Lomp, On semilocal modules and rings, Comm. Algebra 27 (1999), no. 4, 1921-1935. https://doi.org/10.1080/00927879908826539
  18. G. Puninski and P. Rothmaler, When every finitely generated flat module is projective, Journal of Algebra 277 (2004), no. 2, 542-558. https://doi.org/10.1016/j.jalgebra.2003.10.027
  19. J. Xu, Flat covers of modules, Lecture Notes in Mathematics 1634, Springer-Verlag, Berlin, 1996.
  20. W. Xue, Characterizations of semiperfect and perfect rings, Publ. Mat. 40 (1996), no. 1, 115-125. https://doi.org/10.5565/PUBLMAT_40196_08
  21. D. X. Zhou and X. R. Zhang, Small-essential submodules and morita duality, Southeast Asian Bulletin of Mathematics 35 (2011), no. 6, 1051-1062.
  22. Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2000), no. 3, 305-318. https://doi.org/10.1007/s10011-000-0305-9
  23. B. Zimmermann-Huisgen, Direct products of modules and algebraic compactness, Habilitation-schrift, Tech. Univ. Munich, 1980.