Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON ω-LOCAL MODULES AND Rad-SUPPLEMENTED MODULES

  • Buyukasik, Engin (Izmir Institute of Technology Department of Mathematics) ;
  • Tribak, Rachid (Centre Regional des Metiers de l'Education et de la Formation (CRMEF)-Tanger)
  • Received : 2013.10.11
  • Published : 2014.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

All modules considered in this note are over associative commutative rings with an identity element. We show that a ${\omega}$-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that ${\omega}$-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).

Keywords

References

  1. R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. Math. Sci. 2003 (2003), no. 27, 1715-1724. https://doi.org/10.1155/S0161171203202180
  2. F.W. Anderson and K. R. Fuller, Rings and Categories of Modules, New-York, Springer-Verlag, 1974.
  3. E. Buyukasik and C. Lomp, On a recent generalization of semiperfect rings, Bull. Aust. Math. Soc. 78 (2008), no. 2, 317-325. https://doi.org/10.1017/S0004972708000774
  4. E. Buyukasik, E. Mermut, and S. Ozdemir, Rad-supplemented modules, Rend. Semin. Mat. Univ. Padova 124 (2010), 157-177. https://doi.org/10.4171/RSMUP/124-10
  5. J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory. Basel: Frontiers in Mathematics, Birkhauser Verlag, 2006.
  6. S. Ecevit, M. T. Kosan, and R. Tribak. Rad-${\oplus}$-supplemented modules and cofinitely Rad-${\oplus}$-supplemented modules, Algebra Colloq. 19 (2012), no. 4, 637-648. https://doi.org/10.1142/S1005386712000508
  7. A. I. Generalov, The w-cohigh purity in a category of modules, Math. Notes 33 (1983), 402-408; translation from Mat. Zametki 33 (1983), no. 5, 785-796. https://doi.org/10.1007/BF01158292
  8. V. N. Gerasimov and I. I. Sakhaev. A counterexample to two conjectures on projective and flat modules, Sibirsk. Mat. Zh. 25 (1984), no. 6, 31-35. https://doi.org/10.1007/BF00969506
  9. R. W. Gilmer, Multiplicative Ideal Theory, New York, Marcel Dekker, 1972.
  10. R. M. Hamsher, Commutative Noetherian rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 17 (1966), 1471-1472.
  11. S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, Cambridge, UK: London Math. Soc. Lecture Note Ser 147, Cambridge University Press, 1990.
  12. P. Rudlof, On the structure of couniform and complemented modules, J. Pure Appl. Algebra 74 (1991), no. 3, 281-305. https://doi.org/10.1016/0022-4049(91)90118-L
  13. P. Rudlof, On minimax and related modules, Canad. J. Math. 44 (1992), no. 1, 154-166. https://doi.org/10.4153/CJM-1992-009-7
  14. P. F. Smith, Finitely generated supplemented modules are amply supplemented, Arab. J. Sci. Eng. Sect. C Theme Issues 25 (2000), no. 2, 69-79.
  15. R. Tribak, On ${\delta}$-local modules and amply ${\delta}$-supplemented modules, J. Algebra Appl. 12 (2013), no. 2, 1250144, 14 pages.
  16. E. Turkmen and A. Pancar, On cofinitely Rad-supplemented modules, Int. J. Pure Appl. Math. 53 (2009), no. 2, 153-162.
  17. Y.Wang and N. Ding, Generalized supplemented modules, Taiwanese J. Math. 10 (2006), no. 6, 1589-1601. https://doi.org/10.11650/twjm/1500404577
  18. R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), no. 1, 233-256. https://doi.org/10.1090/S0002-9947-1971-0274511-2
  19. R. Wisbauer, Foundations of Module and Ring Theory, Philadelphia: Gordon and Breach Science Publishers, 1991.
  20. H. Zoschinger, Gelfandringe und koabgeschlossene Untermoduln, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 3 (1982), 43-70.
  21. H. Zoschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1-32. https://doi.org/10.1016/0021-8693(86)90125-0

Cited by

  1. RAD-SUPPLEMENTING MODULES vol.53, pp.2, 2016, https://doi.org/10.4134/JKMS.2016.53.2.403