Cohereditary Modules in σ[M]

  • Derya Keskin Tütüncü
  • Nil Orhan Ertaş
  • Rachid Tribak
Part of the Trends in Mathematics book series (TM)


A module N ∈ σ[M] is called cohereditary in σ[M] if every factor module of N is injective in σ[M]. This paper explores the properties and the structure of some classes of cohereditary modules. Among others, we prove that any cohereditary lifting semi-artinian module in σ[M] is a direct sum of Artinian uniserial modules. We show that over a commutative ring a lifting module N with small radical is cohereditary in σ[M] if and only if N is semisimple M-injective. It is also shown that if E is an indecomposable injective module over a commutative Noetherian ring R with associated prime ideal p, then E is cohereditary lifting if and only if there is only one maximal ideal m over p and the ring R m is a discrete valuation ring.


Cohereditary module Lifting module Injective module Semisimple module 


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  1. [1]
    M. Alkan and A. Harmanci, On summand sum and summand intersection property of modules, Turkish J. Math. 26 (2002), 131–147.MathSciNetzbMATHGoogle Scholar
  2. [2]
    F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin, Heidelberg, New York, 1974.zbMATHGoogle Scholar
  3. [3]
    Y. Baba and M. Harada, On almost M-projectives and almost M-injectives, Tsukuba J. Math. 14(1) (1990), 53–69.MathSciNetzbMATHGoogle Scholar
  4. [4]
    N. Bourbaki, Algèbre commutative, Chapitres 5 à 7. Masson, Paris, 1985.Google Scholar
  5. [5]
    J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Frontiers in Mathematics, Birkhäuser, 2006.zbMATHGoogle Scholar
  6. [6]
    N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics series, Longman, Harlow, 1994.Google Scholar
  7. [7]
    L. Ganesan and N. Vanaja, Modules for which every submodule has a unique coclosure, Comm. Algebra 30(5) (2002), 2355–2377.CrossRefMathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Hausen, Supplemented modules over Dedekind domains, Pacific J. Math. 100(2) (1982), 387–402.MathSciNetzbMATHGoogle Scholar
  9. [9]
    A. Idelhadj and R. Tribak, On injective ⊕-supplemented modules, Proceedings of the first Moroccan-Andalusian meeting on algebras and their applications, Tétouan, Morocco, September 2001. Morocco: Université Abdelmalek Essaadi, Faculté des Sciences de Tétouan, Dépt. de Mathématiques et Informatique, UFR-Algèbre et Géométrie Différentielle. (2003), 166–180.Google Scholar
  10. [10]
    A. Idelhadj and R. Tribak, On some properties of ⊕-supplemented modules, Int. J. Math. Math. Sci. 69 (2003), 4373–4387.CrossRefMathSciNetGoogle Scholar
  11. [11]
    A. Idelhadj and R. Tribak, Modules for which every submodule has a supplement that is a direct summand, Arab. J. Sci. Eng. Sect. C Theme Issues 25(2)(2000), 179–189.MathSciNetGoogle Scholar
  12. [12]
    Irving Kaplansky, Infinite Abelian Groups, University of Michigan, 1969.Google Scholar
  13. [13]
    F. Kasch, Modules and Rings, New York: Academic Press, 1982.zbMATHGoogle Scholar
  14. [14]
    D. Keskin, Finite direct sum of(D 1) modules, Turkish J. Math. 22 (1998), 85–91.MathSciNetzbMATHGoogle Scholar
  15. [15]
    T.Y. Lam, Lectures on Modules and Rings, vol. 189 of Graduate Texts in Mathematics, New York: Springer-Verlag, 1998.Google Scholar
  16. [16]
    C. Lomp, On Dual Goldie Dimension, Diplomarbeit (M. Sc. Thesis), University of Düsseldorf, Germany, 1996.Google Scholar
  17. [17]
    S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series, 147, Cambridge, 1990.Google Scholar
  18. [18]
    E. Matlis, 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Mathematics, 327, Springer-Verlag, 1973.Google Scholar
  19. [19]
    B.L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645–650.MathSciNetzbMATHGoogle Scholar
  20. [20]
    K. Oshiro and R. Wisbauer, Modules with every subgenerated module lifting, Osaka J. Math. 32 (1995), 513–519.MathSciNetzbMATHGoogle Scholar
  21. [21]
    D.W. Sharpe and P. Vamos, Injective Modules, Lecture in Pure Mathematics, University of Sheffield, 1972.Google Scholar
  22. [22]
    Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra 30(3) (2002), 1449–1460.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    William D. Weakley, Modules whose proper submodules are finitely generated, J. Algebra 84 (1983), 189–219.CrossRefMathSciNetzbMATHGoogle Scholar
  24. [24]
    R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991.zbMATHGoogle Scholar
  25. [25]
    R. Wisbauer, Tilting in Module Categories, Abelian groups, module theory, and topology (Padua, 1997), Lect. Notes Pure Appl. Math., 201, Marcel Dekker, New York, (1998), 421–444.Google Scholar
  26. [26]
    O. Zarisky and P. Samuel. Commutative Algebra, 1. Springer-Verlag, New York, Heidelberg, Berlin, 1979.Google Scholar
  27. [27]
    H. Zöschinger, Schwach-Injektive Moduln, Period. Math. Hungar. 52(2) (2006), 105–128.CrossRefMathSciNetzbMATHGoogle Scholar
  28. [28]
    H. Zöschinger, Gelfandringe und Koabgeschlossene Untermoduln, Bayer. Akad. Wiss. Math.-Natur. Kl., Sitzungsber., 3 (1982), 43–70.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Derya Keskin Tütüncü
    • 1
  • Nil Orhan Ertaş
    • 2
  • Rachid Tribak
    • 3
  1. 1.Department of MathematicsHacettepe UniversityBeytepe AnkaraTurkey
  2. 2.Department of MathematicsSüleyman Demirel UniversityCünür IspartaTurkey
  3. 3.Département de mathématiquesFaculté des sciences de TétouanTétouanMorocco

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