Skip to main content
SpringerLink
Log in

Modules with Chain Condition on Non-finitely Generated Submodules

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this article, we study modules with chain condition on non-finitely generated submodules. We show that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then M has Noetherian dimension and its Noetherian dimension is less than or equal to one. In particular, we observe that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then every submodule of M is countably generated. We investigate that if an R-module M satisfies the descending chain condition on non-finitely generated submodules, then M has Krull dimension and its Krull dimension may be any ordinal number \(\alpha \). In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albu, T., Rizvi, S.: Chain conditions on quotient finite dimensional modules. Commun. Algebra 29(5), 1909–1928 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albu, T., Smith, P.F.: Duall relative Krull dimension of modules over commutative rings. In: Facchini, A., Menini, C. (eds.) Abelian Groups and Modules, vol. 343, pp. 1–15. Kluwer, Dordrecht (1995)

    Google Scholar 

  3. Albu, T., Smith, P.F.: Dual Krull dimension and duality. Rocky Mt. J. Math. 29, 1153–1164 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Albu, T., Smith, P.F.: Localization of modular lattices, Krull dimension and the Hopkins–Levitzki Theorem(I). Math. Proc. Camb. Philos. Soc. 120, 87–101 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Albu, T., Smith, P.F.: Localization of modular lattices, Krull dimension, and the Hopkins–Levitzki Theorem (II). Commun. Algebra 25, 1111–1128 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Albu, T., Teply, L.: Generalized deviation of posets and modular lattices. Discrete Math. 214, 1–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Albu, T., Vamos, P.: Global Krull dimension and global dual Krull dimension of valuation rings, abelian groups, modules theory, and topology. In: Proc. Marcel-Dekker, pp. 37–54 (1998)

  8. Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  9. Chambless, L.: N-Dimension and N-critical modules, application to artinian modules. Commun. Algebra 8, 1561–1592 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davoudian, M.: Modules satisfying double chain condition on non-finitely generated submodules have Krull dimension. Turk. J. Math. 41, 1570–1578 (2017)

    Article  MathSciNet  Google Scholar 

  11. Davoudian, M., Karamzadeh, O.A.S., Shirali, N.: On \(\alpha \)-short modules. Math. Scand. 114(1), 26–37 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gordon, R.: Gabriel and Krull dimension. In: Ring Theory, Proceeding of the Oklahoma Conference Lecture Notes in Pure and Applied Mathematics, vol. 7. Dekker, New York, pp. 241–295 (1974)

  13. Gordon, R., Robson, J.C.: Krull dimension. Mem. Am. Math. Soc. 133 (1973)

  14. Hashemi, J., Karamzadeh, O.A.S., Shirali, N.: Rings over which the Krull dimension and Noetherian dimension of all modules coincide. Commun. Algebra 37, 650–662 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Karamzadeh, O.A.S., Motamedi, M.: On \(\alpha \)-\(Dicc\) modules. Commun. Algebra 22, 1933–1944 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karamzadeh, O.A.S.: Noetherian-dimension, Ph.D. thesis. Exeter (1974)

  17. Karamzadeh, O.A.S., Rahimpur, Sh: \(\lambda \)-finitely embeded modules. Algebra Colloq. 12, 281–292 (2005)

    Article  MathSciNet  Google Scholar 

  18. Karamzadeh, O.A.S., Sajedinejad, A.R.: Atomic modules. Commun. Algebra 29(7), 2757–2773 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karamzadeh, O.A.S., Sajedinejad, A.R.: On the Loewy length and the Noetherian dimension of Artinian modules. Commun. Algebra 30, 1077–1084 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Karamzadeh, O.A.S., Shirali, N.: On the countablity of Noetherian dimension of modules. Commun. Algebra 32, 4073–4083 (2004)

    Article  MATH  Google Scholar 

  21. Krause, G.: On the Krull-dimension of left Noetherian rings. J. Algebra 23, 88–99 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kirby, D.: Dimension and length for Artinian modules. Q. J. Math. Oxf. 41, 419–429 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lemonnier, B.: Déviation des ensembles et groupes abéliens totalement ordonnés. Bull. Sci. Math. 96, 289–303 (1972)

    MathSciNet  MATH  Google Scholar 

  24. Lemonnier, B.: Dimension de Krull et codeviation Application au theorem d’Eakin. Commun. Algebra 16, 1647–1665 (1978)

    Article  MATH  Google Scholar 

  25. McConell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley-Interscience, New York (1987)

    Google Scholar 

  26. Roberts, R.N.: Krull-dimension for Artinian modules over quasi local commutative rings. Q. J. Math. Oxf. 26, 269–273 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vamos, P.: The dual of the notion of finitely generated. J. Lond. Math. Soc. 43, 642–646 (1969)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the well-informed referee of this article for the detailed report, corrections, and several constructive suggestions for improvement.

Author information

Authors and Affiliations

  1. Department of mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Maryam Davoudian

Authors
  1. Maryam Davoudian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maryam Davoudian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davoudian, M. Modules with Chain Condition on Non-finitely Generated Submodules. Mediterr. J. Math. 15, 1 (2018). https://doi.org/10.1007/s00009-017-1047-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-1047-y

Keywords

Mathematics Subject Classification

Search

Navigation