Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Some Characterizations of Modules via Essentially Small Submodules

  • Le, Van Thuyet (Department of Mathematics, Hue University) ;
  • Phan, Hong Tin (Department of Mathematics, Hue University's College of Education)
  • Received : 2015.03.12
  • Accepted : 2016.05.11
  • Published : 2016.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the structure of e-local modules and classes of modules via essentially small are investigated. We show that the following conditions are equivalent for a module M: (1) M is e-local; (2) $Rad_e(M)$ is a maximal submodule of M and every proper essential submodule of M is contained in a maximal submodule; (3) M has a unique essential maximal submodule and every proper essential submodule of M is contained in a maximal submodule.

Keywords

References

  1. I. Al-Khazzi and P. F. Smith, Modules with chain conditions on super uous submod-ules, Comm. Algebra, 19(8)(1991), 2331-2351.
  2. F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Springer-Verlag - New York, 1974.
  3. P. Aydogdu, Rings over which every module has a flat ${\delta}$-cover, Turkish J. Math., 37(2013), 182-194.
  4. E. Buyukasik and C. Lomp, When -semiperfect rings are semiperfect, Turkish J. Math., 34(2010), 317-324.
  5. N. Er, E. Nil Orhan, Lifting modules with indecomposable decompositions, Comm. Algebra, 36(2)(2008), 395-404. https://doi.org/10.1080/00927870701715738
  6. R. E. Jonhson, Structure theory of faithful rings, Trans. Amer. Math. Soc., 84(1957), 508-522.
  7. M. T. Kosan, ${\delta}$-lifting and ${\delta}$-supplemented modules, Algebra Colloq., 14(1)(2007), 53-60. https://doi.org/10.1142/S1005386707000065
  8. D. Keskin Tutuncu and R. Tribak, On dual Baer modules, Glassgow Math. J., 52(2010), 261-269. https://doi.org/10.1017/S0017089509990334
  9. D. Keskin Tutuncu and R. Tribak, On $\tau$-noncosingular modules, Bull. Aust. Math. Soc., 80(2009), 462-471. https://doi.org/10.1017/S0004972709000409
  10. A. C. Ozcan, The torsion theory cogenerated by ${\delta}$-M-small modules and GCO-modules, Comm. Algebra, 35(2007), 623-633. https://doi.org/10.1080/00927870601074871
  11. T.C. Quynh, On pseudo semi-projective modules, Turkish J. Math., 37(1)(2013), 27-36.
  12. T. C. Quynh and P. H. Tin, Somes Properties of e-supplemented and e-lifting modules, Vietnam J. Math., 41(3)(2013), 303-312. https://doi.org/10.1007/s10013-013-0022-6
  13. Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra, 30(3)(2002), 1449-1460. https://doi.org/10.1080/00927870209342390
  14. R. Tribak, Some results on $\tau$-noncosingular modules, Turkish J. Math., 38(2014), 29-39. https://doi.org/10.3906/mat-1302-52
  15. R. Tribak, On -local modules and amply ${\delta}$-supplemented modules, J. Algebra Appl., 12(2)(2013), 1250144, 14 pp.
  16. Y. Wang, ${\delta}$-small submodules and ${\delta}$-supplemented Modules, Int. J. Math. Math. Sci. (2007), 58-132
  17. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading 1991.
  18. M. Yousif and Y. Zhou, Semiregular, Semiperfect and perfect rings relative to an ideal, Rocky Mountain J. Math., 32(4)(2002) 1651-1671. https://doi.org/10.1216/rmjm/1181070046
  19. Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq., 7(2000), 305-318. https://doi.org/10.1007/s10011-000-0305-9
  20. D. X. Zhou, X. R. Zhang, Small-essential submodules and Morita duality, Southeast Asian Bull. Math., 35(2011), 1051-1062.