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On Regular Modules over Commutative Rings

  • Dawood Hassanzadeh-LelekaamiEmail author
  • Hajar Roshan-Shekalgourabi
Article

Abstract

In this paper, we investigate the class of von Neumann regular modules over commutative rings. More precisely, we introduce a characterization of regular modules, and then, we study some properties of these modules in viewpoint of this characterization. Among other things, we show that the Nakayama’s Lemma and Krull’s intersection theorem hold for this class of modules. Also, some explicit expressions for submodules of regular modules are introduced.

Keywords

Von Neumann regular ring Regular module Semisimple module Krull’s intersection theorem Prime submodule 

Mathematics Subject Classification

16E50 13C13 13C05 16D40 16D60 13C11 

Notes

Acknowledgements

The authors specially thank the referee for the helpful suggestions and comments.

References

  1. 1.
    Azizi, A.: Prime submodules of Artinian modules. Taiwan. J. Math. 13(6B), 2011–2020 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bruns, W., Herzog, J.: Cohen–Macaulay Rings, revised edn. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. 3.
    Bourbaki, N.: Commutative Algebra, Chapters 1–7. Hermann, Paris (1972)Google Scholar
  4. 4.
    Cheatham, F.D.: F-absolutely pure modules. Ph.D. dissertation. University of Kentucky (1972)Google Scholar
  5. 5.
    Fieldhouse, D.J.: Pure theories. Math. Ann. 184, 1–18 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fieldhouse, D. J.: Regular modules over semi-local rings, Rings, modules and radicals (proc. Colloq., Keszthely, 1971), 193–196. Colloq. Math. Soc. Janos Bolyai, 6, North-Holland, Amsterdam (1973)Google Scholar
  7. 7.
    Goodearl, K.R.: Von Neumann regular rings monographs and studies in mathematics, vol. 4. Pitman Publishing Pty Ltd, Melbourne (1979)Google Scholar
  8. 8.
    Gheatham, T.J., Smith, J.R.: Regular and semisimple modules. Pac. J. Math. 65(2), 315–323 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kaplansky, I.: Projective modules. Ann. Math. Second Ser. 68(2), 372–377 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaplansky, I.: Commutative Rings. University of Chicago Press, Chicago (1974)zbMATHGoogle Scholar
  11. 11.
    Lambek, J.: Lectures on Rings and Modules. Blaisdell, Waltham, Mass (1966)zbMATHGoogle Scholar
  12. 12.
    Lu, C.-P.: Prime submodules of modules. Comment. Math. Univ. St. Pauli 33(1), 61–69 (1984)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lu, C.-P.: M-radicals of submodules in modules. Math. Jpn. 34(2), 211–219 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lu, C.-P.: Spectra of modules. Commun. Algebra 23(10), 3741–3752 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lu, C.-P.: Saturations of submodules. Commun. Algebra 31(6), 2655–2673 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McCoy, N.H.: Rings and Ideals, Carus Monograph. Mathematical Association of America, Washington (1978)Google Scholar
  17. 17.
    McCasland, R.L., Moore, M.E.: On radicals of submodules. Commun. Algebra 19, 1327–1341 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    McCasland, R.L., Moore, M.E.: Prime submodules. Commun. Algebra 20(6), 1803–1817 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    McCasland, R.L., Smith, P.F.: Prime submodules of Noetherian modules. Rocky Mt. J. Math. 23(3), 1041–1062 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ramamurthi, V.S., Rangaswamy, K.M.: On finitely injective modules. J. Aust. Math. Soc. 163(16), 239–248 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ware, R.: Endomorphism rings of projective modules. Trans. Am. Math. Soc. 155, 233–256 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wisbauer, R.: Foundations of Modules and Ring Theory. Gordon and Breach Reading, Reading (1991)zbMATHGoogle Scholar
  23. 23.
    Zelmanowitz, J.: Regular modules. Trans. Am. Math. Soc. 163, 341–355 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  • Dawood Hassanzadeh-Lelekaami
    • 1
    Email author
  • Hajar Roshan-Shekalgourabi
    • 1
  1. 1.Department of Basic SciencesArak University of TechnologyArakIran

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