Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 2, pp 429–445 | Cite as

Retractable and Coretractable Modules over Formal Triangular Matrix Rings

  • Derya Keskin Tütüncü
  • Rachid TribakEmail author
Original Paper


In this paper, we study retractable modules and coretractable modules over a formal triangular matrix ring \(T=\left[ \begin{array}{rr} A &{} 0 \\ M &{} B \\ \end{array} \right] \), where A and B are rings and M is a (BA)-bimodule. We determine necessary and sufficient conditions for a T-module to be, respectively, retractable or coretractable. We also characterize the right Kasch formal triangular matrix rings. Some examples are provided to illustrate and delimit our results.


Coretractable modules Formal triangular matrix rings Right Kasch rings Retractable modules 

Mathematics Subject Classification

Primary 16D10 Secondary 16D20 16D70 16D80 16S50 


  1. 1.
    Abyzov, A.N., Tuganbaev, A.A.: Retractable and coretractable modules. J. Math. Sci. (N.Y.) 213(2), 132–142 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amini, B., Ershad, M., Sharif, H.: Coretractable modules. J. Aust. Math. Soc. 86, 289–304 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, New York (1974)CrossRefzbMATHGoogle Scholar
  4. 4.
    Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics. Birkhäuser, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Ecevit, Ş., Koşan, M.T.: On rings all of whose modules are retractable. Arch. Math. (Brno) 45, 71–74 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Faith, C.: Rings whose modules have maximal submodules. Publ. Mat. 39, 201–214 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goodearl, K.R.: Ring Theory, Nonsingular Rings and Modules. Marcel Dekker, New York (1976)zbMATHGoogle Scholar
  8. 8.
    Green, E.L.: On the representation theory of rings in matrix form. Pacific J. Math. 100(1), 123–138 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haghany, A.: Injectivity conditions over a formal triangular matrix ring. Arch. Math. (Basel) 78, 268–274 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Haghany, A., Varadarajan, K.: Study of formal triangular matrix rings. Commun. Algebra 27(11), 5507–5525 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haghany, A., Varadarajan, K.: Study of modules over formal triangular matrix rings. J. Pure Appl. Algebra 147, 41–58 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Haghany, A., Karamzadeh, O.A.S., Vedadi, M.R.: Rings with all finitely generated modules retractable. Bull. Iranian Math. Soc. 35(2), 37–45 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Haghany, A., Mazrooei, M., Vedadi, M.R.: Pure projectivity and pure injectivity over formal triangular matrix rings. J. Algebra Appl. 11(6), 1250107 (2012). (13 pages)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Keskin Tütüncü, D., Kalebog̃az, B.: On coretractable modules. Hokkaido Math. J. 44(1), 91–99 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Khuri, S.M.: Nonsingular retractable modules and their endomorphism rings. Bull. Aust. Math. Soc. 43(1), 63–71 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krylov, P.A., Tuganbaev, A.A.: Modules over formal matrix rings. J. Math. Sci. (N.Y.) 171(2), 248–295 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lam, T.Y.: Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189. Springer, New York (1999)CrossRefGoogle Scholar
  18. 18.
    Dung, N.V., Smith, P.F.: On semi-artinian \(V\)-modules. J. Pure Appl. Algebra 82(1), 27–37 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nguyen, D.V., Dinh, H.V., Smith, P.F., Wisbauer, R.: Extending Modules. Longman Scientific and Technical, New York (1994)zbMATHGoogle Scholar
  20. 20.
    Nicholson, W.K., Yousif, M.F.: On perfect simple-injective rings. Proc. Am. Math. Soc. 125(4), 979–985 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nicholson, W.K., Yousif, M.F.: Quasi-Frobenius Rings, vol. 158. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Osofsky, B.L.: A generalization of quasi-Frobenius rings. J. Algebra 4, 373–387 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Smith, P.F.: Modules with many homomorphisms. J. Pure Appl. Algebra 197, 305–321 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach, Philadelphia (1991)zbMATHGoogle Scholar
  25. 25.
    Zemlicka, J.: Completely coretractable rings. Bull. Iranian Math. Soc. 39(3), 523–528 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsHacettepe UniversityAnkaraTurkey
  2. 2.Centre Régional des Métiers de L’Education et de la Formation (CRMEF)-TangerTangierMorocco

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