The Krull Dimension of Certain Semiprime Modules Versus Their \(\alpha \)-Shortness

  • S. M. Javdannezhad
  • N. Shirali


We study the R-modules M which are finitely generated, quasi-projective and self-generator (briefly called \({{\mathrm{FQS}}}\) modules). We extend some basic results from semiprime rings to semiprime \({{\mathrm{FQS}}}\) modules. In particular, we show that any semiprime \({{\mathrm{FQS}}}\) module with Krull dimension is a Goldie module. We also show that every \({{\mathrm{FQS}}}\) module with Krull dimension has only finitely many minimal prime submodules. Consequently, if M is an \({{\mathrm{FQS}}}\) module with Krull dimension, then \(\text{ k-dim }\,M\) is equal to \(\text{ k-dim }\,\frac{M}{P}\) for some prime submodule P of M. Moreover, we observe that an \({{\mathrm{FQS}}}\) module has the classical Krull dimension if and only if it satisfies ACC on prime submodules. Finally, we prove that a semiprime \({{\mathrm{FQS}}}\) module M is \(\alpha \)-short if and only if \(\text{ n-dim }\, M =\alpha ,\) where \(\alpha \ge 0.\)


\({{\mathrm{FQS}}}\) module Krull dimension classical Krull dimension 

Mathematics Subject Classification

Primary 16P60 16P20 Secondary 16P40 


  1. 1.
    Albu, T.: Sur la dimension de Gabriel des modules, Algebra-Berichte, Bericht Nr. 21, Seminar F. Kasch-B. Pareigis, Mathematisches Institut der Universitat München, Verlag Uni Druck (1974)Google Scholar
  2. 2.
    Albu, T., Rizvi, S.: Chain conditions on quotient finite dimensional modules. Commun. Algebra 29(5), 1909–1928 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Albu, T., Smith, P.F.: Dual Krull dimension and duality. Rocky Mt. J. Math. 29, 1153–1165 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Albu, T., Teply, L.: Generalized deviation of posets and modular lattices. Discret. Math. 214, 1–19 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Albu, T., Vamos, P.: Global Krull Dimension and Global Dual Krull Dimension of Valuation Rings. Lecture Notes in Pure and Applied Mathematics, vol. 201, pp. 37–54 (1998)Google Scholar
  7. 7.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Graduate Texts in Mathematics, vol. 13. Springer, Berlin (1992)CrossRefGoogle Scholar
  8. 8.
    Behboodi, M., Karamzadeh, O.A.S., Koohy, H.: Modules whose certain submodules are prime. Vietnam J. Math. 32, 303–317 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bilhan, G., Smith, P.F.: Short modules and almost Noetherian modules. Math. Scand. 98, 12–18 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Contessa, M.: On modules with DICC. J. Algebra 107, 75–81 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Contessa, M.: On DICC rings. J. Algebra 105, 429–436 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Contessa, M.: On rings and modules with DICC. J. Algebra 101, 489–496 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davoudian, M., Karamzadeh, O.A.S., Shirali, N.: On \(\alpha \)-short modules. Math. Scand. 114, 26–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dauns, J.: Prime modules. J. Reine Angew. Math. 298, 156–181 (1978)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R.: Extending Modules. Pitman Research Notes in Mathematics Series, vol. 313. Longman Scientific and Technical, Harlow (1994)Google Scholar
  16. 16.
    Garcia, J.L., Gomez Pardo, J.L.: On endomorphism rings of quasiprojective modules. Math. Z. 196, 87–108 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goodearl, K.R., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  18. 18.
    Gordon, R., Robson, J.C.: Krull dimension. Mem. Am. Math. Soc. 133 (1973)Google Scholar
  19. 19.
    Karamzadeh, O.A.S., Motamedi, M.: On \(\alpha \)-DICC modules. Commun. Algebra 22, 1933–1944 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koehler, A.: Quasi-projective covers and direct sums. Am. Math. Soc. 24, 655–658 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Krause, G.: On fully left bounded left Noetherian rings. J. Algebra 23, 88–99 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lemonnier, B.: Dimension de Krull et codeviation, Application au theorem dÉakin. Commun. Algebra 6, 1647–1665 (1978)CrossRefzbMATHGoogle Scholar
  23. 23.
    Lemonnier, B.: Déviation des ensembles et groupes abéliens totalement ordonnés. Bull. Sci. Math. 96, 289–303 (1972)MathSciNetzbMATHGoogle Scholar
  24. 24.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley-Interscience, New York (1987)zbMATHGoogle Scholar
  25. 25.
    McCasland, R.L., Smith, P.F.: Prime submodules of Noetherian modules. Rocky Mt. J. Math. 23, 1041–1062 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sanh, N.V., Asawasamrit, S., Ahmed, K.F.U., Thao, L.P.: On prime and semiprime Goldie modules. Asian Eur. J. Math. 4(2), 321–334 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sanh, N.V., Vu, N.A., Ahmed, K.F.U., Asawasamrit, S., Thao, L.P.: Primeness in module category. Asian Eur. J. Math. 3(1), 145–154 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach Science Publishers, Reading (1991)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShahid Chamran University of AhvazAhvazIran

Personalised recommendations