Advertisement

Dual CS-Rickart Modules over Dedekind Domains

  • Rachid Tribak
Article
  • 4 Downloads

Abstract

We study d-CS-Rickart modules (i.e. modules M such that for every endomorphism φ of M, the image of φ lies above a direct summand of M) over Dedekind domains. The structure of d-CS-Rickart modules over discrete valuation rings is fully determined. It is also shown that for a d-CS-Rickart R-module M over a nonlocal Dedekind domain R, the following assertions hold:
  1. (i)

    The \(\mathfrak {p}\)-primary component of M is a direct summand of M for any nonzero prime ideal \(\mathfrak {p}\) of R.

     
  2. (ii)

    M/T(M) is an injective R-module, where T(M) is the torsion submodule of M.

     
  3. (iii)

    If, moreover, M is a reduced R-module, then \(\bigoplus _{\mathfrak {p} \in \mathbf {P}} T_{\mathfrak {p}}(M) \leq M \leq {\prod }_{\mathfrak {p} \in \mathbf {P}} T_{\mathfrak {p}}(M),\) where P is the set of all nonzero prime ideals of R and \(T_{\mathfrak {p}}(M)\) is the \(\mathfrak {p}\)-primary component of M for every \(\mathfrak {p} \in \mathbf {P}\).

     

Keywords

Endomorphisms d-CS-Rickart modules d-Rickart modules Lifting modules 

Mathematics Subject Classification (2010)

16D10 16D70 16D80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author is very grateful to the referee for valuable suggestions and comments which improved this paper.

References

  1. 1.
    Abyzov, A.N., Nhan, T.H.N.: CS-Rickart modules. Russian mathematics (Iz. VUZ) 58(5), 48–52 (2014). Original Russian Text published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematica, 5 (2014) 59–63MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abyzov, A.N., Nhan, T.H.N.: CS-Rickart modules. Lobachevskii J. Math. 35 (4), 317–326 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, Graduate Texts in Mathematics, vol. 13. Springer, New York (1974)CrossRefGoogle Scholar
  4. 4.
    Armendariz, E.P.: A note on extensions of Baer and P.P.-rings. J. Austral. Math. Soc. 18, 470–473 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, London (1969)zbMATHGoogle Scholar
  6. 6.
    Berberian, S.K.: Baer *-Rings, Die Grundlehren der mathematischen Wissenschaften Band, vol. 195. Springer, New York (1972)Google Scholar
  7. 7.
    Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics. Basel, Birkhäuser (2006)zbMATHGoogle Scholar
  8. 8.
    Dung, N.V.: Modules with indecomposable decompositions that complement maximal direct summands. J. Algebra 197, 449–467 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R.: Extending Modules, Pitman Research Notes in Mathematics Series, vol. 313. Longman, Harlow (1994)Google Scholar
  10. 10.
    Endo, S.: Note on pp. rings. Nagoya Math. J. 17, 167–170 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fuchs, L.: Infinite Abelian Groups, vol. II. Academic Press, New York (1973)zbMATHGoogle Scholar
  12. 12.
    Generalov, A.I.: The ω-cohigh purity in a category of modules. Math. Notes 33(5), 402–408 (1983). translation from Mat. Zametki 33(5) (1983) 785–796MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hattori, A.: A foundation of torsion theory for modules over general rings. Nagoya Math. J. 17, 147–158 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hausen, J.: Supplemented modules over Dedekind domains. Pacific J. Math. 100 (2), 387–402 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplansky, I.: Modules over Dedekind rings and valuation rings. Trans. Amer. Math. Soc. 72, 327–340 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, G., Rizvi, S.T., Roman, C.S.: Dual Rickart modules. Comm. Algebra 39(11), 4036–4058 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, W., Chen, J.: When CF rings are artinian. J. Algebra Appl. 12(4) (2013).  https://doi.org/10.1142/S0219498812500594
  18. 18.
    Maeda, S.: On a ring whose principal right ideals generated by idempotenets form a lattice. J. Sci. Hiroshima Univ. Ser. A 24, 509–525 (1960)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mohamed, S.H., Müller, B.J.: Continuous and Discrete Modules. London Math. Soc Lecture Note Series, vol. 147. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  20. 20.
    Nicholson, W.K.: Semiregular modules and rings. Can. J. Math. 18(5), 1105–1120 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nicholson, W.K., Yousif, M.F.: Weakly continuous and C 2-rings. Comm. Algebra 29(6), 2429–2446 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Özcan, A. Ç., Harmanci, A.: Duo modules. Glasgow Math. J. 48, 533–545 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rizvi, S.T., Roman, C.S.: Baer Property of Modules and Applications, Advances in Ring Theory, pp 225–241. World Sci. Publ., Hackensack (2005)zbMATHGoogle Scholar
  24. 24.
    Sharpe, D.W., Vámos, P.: Injective Modules. Cambridge University Press, Cambridge (1972)zbMATHGoogle Scholar
  25. 25.
    Singh, S.: Semi-dual continuous modules over Dedekind domains. J. Univ. Kuwait (Sci.) 11, 33–39 (1984)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Tuganbaev, A.: Rings Close to Regular, Mathematics and Its Applications, vol. 545. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
  27. 27.
    Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach Science Publishers, Philadelphia (1991)zbMATHGoogle Scholar
  28. 28.
    Zöschinger, H.: Koatomare moduln. Math. Z. 170, 221–232 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zöschinger, H.: Komplemente als direkte summanden. Arch. Math. (Basel) 25, 241–253 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zöschinger, H.: Komplemente als direkte summanden II. Arch. Math. (Basel) 38, 324–334 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.(CRMEF)-TangerCentre Régional des Métiers de L’Education et de la Formation (CRMEFTTH)TangierMorocco

Personalised recommendations