Dimension of Non-small Submodules

  • Maryam DavoudianEmail author
Original Paper


In this article, we introduce and study the concepts of non-small Krull dimension and non-small Noetherian dimension of an R-module M, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension of modules. They, respectively, rely on the behavior of descending and ascending chains of non-small submodules. We show that, if the non-small Noetherian dimension (resp., non-small Krull dimension) of an R-module M equals \(\alpha \), then \(\frac{M}{\mathrm{Rad}(M)}\) has Noetherian dimension (resp., Krull dimension) and its Noetherian dimension (resp., Krull dimension) is less than or equal to \(\alpha \) (resp., \( \alpha +1\)). For decomposable modules, we show that the non-small Krull dimension (resp., non-small Noetherian dimension) and the Krull dimension (resp., Noetherian dimension) coincide.


Noetherian dimension Small submodule Krull dimension Non-small Krull dimension Non-small Noetherian dimension 

Mathematics subject classification

16P60 16P20 16P40 



The author would like to thank professor O.A.S. Karamzadeh for a useful discussion and his encouragement during the preparation of a revised version of this article. She would also like to thank the referee for carefully reading the paper, giving a detailed report with very helpful comments.


  1. 1.
    Albu, T., Rizvi, S.: Chain conditions on quotient finite dimensional modules. Commun. Algebra 29(5), 1909–1928 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albu, T., Smith, P.F.: Dual Krull dimension and duality. Rocky Mt. J. Math. 29, 1153–1164 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Albu, T., Smith, P.F.: Duall Relative Krull Dimension of Modules over Commutative Rings. In: Facchini, A., Menini, C. (eds.) Abelian Groups and Modules, pp. 1–15. Kluwer, Dordrecht (1995)Google Scholar
  4. 4.
    Albu, T., Vamos, P.: Global Krull dimension and global dual Krull dimension of valuation rings, abelian groups, modules theory, and topology. Lect. Notes Pure Appl. Math. 201, 37–54 (1998)zbMATHGoogle Scholar
  5. 5.
    Al-Khazzi, I., Smith, P.F.: Modules with chain condition on supperfluous submodules. Commun. Algebra 19(8), 2332–2351 (1991)CrossRefGoogle Scholar
  6. 6.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chambless, L.: N-dimension and N-critical modules, application to Artinian modules. Commun. Algebra 8(16), 1561–1592 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davoudian, M.: Dimension of non-finitely generated submodules. Vietnam J. Math. 44, 817–827 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davoudian, M., Karamzadeh, O.A.S., Shirali, N.: On \(\alpha \)-short modules. Math. Scand. 114(1), 26–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davoudian, M.: Modules satisfying double chain condition on non-finitely generated submodules have Krull dimension. Turk. J. Math. 41, 1570–1578 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Davoudian, M., Karamzadeh, O.A.S.: Artinian serial modules over commutative (or left Noetherian) rings are at most one step away from being Noetherian. Commun. Algebra 44, 3907–3917 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davoudian, M., Halali, A., Shirali, N.: On \(\alpha \)-almost Artinian modules. Open Math. 14, 404–413 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davoudian, M., Shirali, N.: On \(\alpha \)-Tall modules. Bull. Malays. Math. Sci. Soc. 41(4), 1739–1747 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Davoudian, M.: On \(\alpha \)-quasi short modules. Int. Electron. J. Algebra 21(1), 91–102 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Davoudian, M., Ghayour, O.: The length of Artinian modules with countable Noetherian dimension. Bull. Iran. Math. Soc. 43(6), 1621–1628 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Davoudian, M.: Modules with chain condition on non-finitely generated submodules. Mediterr. J. Math. 15(1), 1–12 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Davoudian, M.: Dimension on non-essential submodules. J. Algebra Appl. (2019).
  18. 18.
    Gordon, R., Robson, J.C.: Krull dimension. Memoirs of the American Mathematical Society, vol. 133. American Mathematical Society, Providence, R.I. (1973)Google Scholar
  19. 19.
    Lomp, C.: On dual Goldie dimension, Ph.D. thesis, Dusseldorf (1996)Google Scholar
  20. 20.
    Karamzadeh, O.A.S.: Noetherian-dimension, Ph.D. thesis, Exeter (1974)Google Scholar
  21. 21.
    Karamzadeh, O.A.S., Motamedi, M.: On \(\alpha \)-\(Dicc\) modules. Commun. Algebra 22, 1933–1944 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Karamzadeh, O.A.S., Sajedinejad, A.R.: Atomic modules. Commun. Algebra 29(7), 2757–2773 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kirby, D.: Dimension and length for Artinian modules. Q. J. Math. Oxf. 41(2), 419–429 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lemonnier, B.: Deviation des ensembless et groupes totalement ordonnes. Bull. Sci. Math. 96, 289–303 (1972)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lemonnier, B.: Dimension de Krull et codeviation, application au théorèm d’Eakin. Commun. Algebra 6, 1647–1665 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    McConell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley-Interscience, New York (1987)Google Scholar
  27. 27.
    Smith, P.F., Vedadi, M.R.: Modules with chain condition on non-essential submodules. Commun. Algebra 32(5), 1881–1894 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsShahid Chamran University of AhvazAhvazIran

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