Algebras and Representation Theory

, Volume 21, Issue 6, pp 1333–1342 | Cite as

Several Generalizations of the Wedderburn-Artin Theorem with Applications

  • M. BehboodiEmail author
  • A. Daneshvar
  • M. R. Vedadi


We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) \(R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})\) where k,n 1,…,n k and each D i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m 1 ⊕… ⊕ R m k where each R m i is either a simple R-module or a virtually simple direct summand of R.


Virtually simple module Virtually semisimple module Principal left V-domain Wedderburn-Artin Theorem Krull-Schmidt Theorem 

Mathematics Subject Classification (2010)

Primary: 16D60 16D70 16K99 Secondary: 16S50 15B33 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors owe a great debt to the referee who has carefully read earlier versions of this paper and made significant suggestions for improvement. We would like to express our deep appreciation for the referee’s work.


  1. 1.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, Second Edition Graduate Texts in Mathematics, vol. 13. Springer-Verlag, New York (1992)CrossRefGoogle Scholar
  2. 2.
    Artin, E.: Zur Theorie der hyperkomplexen Zahlen. Abh. Math. Sem. U. Hamburg, 5, 251–260 (1927)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Behboodi, M., Behboodi Eskandari, G.: Local duo rings whose finitely generated modules are direct sums of cyclics. Indian J. Pure Appl. Math. 46(1), 59–72 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Behboodi, M., Behboodi Eskandari, G.: On rings over which every finitely generated module is a direct sum of cyclic modules. Hacettepe J. Mathematics and Statistics 45(5), 1335–1342 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Behboodi, M., Daneshvar, A., Vedadi, M.R.: Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem, Comm. Algebra, (to appear). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Behboodi, M., Ghorbani, A., Moradzadeh-Dehkordi, A., Shojaee, S.H.: On left köthe rings and a generalization of a köthe-cohen-kaplansky theorem. Proc. Amer. Math. Soc. 142(8), 2625–2631 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brandal, W.: Commutative Rings Whose Finitely Generated Modules Decompose, Lecture Notes in Mathematics, vol. 723. Springer, Berlin (1979)CrossRefGoogle Scholar
  8. 8.
    Camillo, V.P., Cozzens, J.: A theorem on Noetherian hereditary rings. Pacific J. Math. 45, 35–41 (1973)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cohen, I.S., Kaplansky, I.: Rings for which every module is a direct sum of cyclic modules. Math. Z. 54, 97–101 (1951)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cohn, P.M.: Free Ideal Rings and Localization in General Rings, vol. 3. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  11. 11.
    Clark, J., Huynh, D.V.: A study of uniform one-sided ideals in simple rings. Glasg Math. J. 49, 489–495 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R.: Extending Modules, vol. 313. CRC Press, Florida (1994)Google Scholar
  13. 13.
    Gilmer, R.: Commutative rings in which each prime ideal is principal. Math. Ann. 183, 151–158 (1969)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goodearl, R.K., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings, Second Edition London Mathematical Society Student Texts, vol. 61. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  15. 15.
    Gordon, R., Robson, J.C.: Krull Dimension, Memoirs of the American Mathematical Society, No. 133. American Mathematical Society, Providence (1973)Google Scholar
  16. 16.
    Jain, S.K., Ashish, K.S., Askar, A.T.: Cyclic modules and the structure of rings. Oxford University Press, Oxford (2012)CrossRefGoogle Scholar
  17. 17.
    Jain, S.K., Lam, T.Y., Leroy, A.: Ore Extensions and V-domains, Rings, Modules and Representations, 249–262, Contemp. Math., 480, Amer. Math. Soc., Providence, RI (2009)Google Scholar
  18. 18.
    Köthe, G.: Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring. Math. Z. 39, 31–44 (1935)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lam, T.Y.: Lectures on Modules and Rings Graduate Texts in Mathematics, vol. 189. Springer-Verlag, New York (1999)CrossRefGoogle Scholar
  20. 20.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings, With the Cooperation of L. W. Small, Revised Edition Graduate Studies in Mathematics, vol. 30. American Mathematical Society, Providence (2001)Google Scholar
  21. 21.
    Sabinin, L., Sbitneva, L., Shestakov, I.: Non-associative algebra and its applications, Lecture Notes in Pure and Applied Mathematics 246 Chapman and hall/CRC (2006)Google Scholar
  22. 22.
    Wedderburn, J.H.M.: On hypercomplex numbers. Proc. London Math. Soc. Ser. 2(6), 77–118 (1907)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Wiegand, R., Wiegand, S.: Commutative Rings Whose Finitely Generated Modules are Direct Sums of Cyclics, Abelian Group Theory (Proceedings Second New Mexico State University Conf., Las Cruces, N.M., 1976), Pp. 406-423 Lecture Notes in Math, vol. 616. Springer, Berlin (1977)Google Scholar
  24. 24.
    Wisbauer, R.: Foundations of Module and Ring Theory, A Handbook for Study and Research, Revised and Translated from the 1988 German Edition. Algebra Logic and Applications, vol. 3. Gordon and Breach Science Publishers, Philadelphia (1991)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

Personalised recommendations