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QI-modules

  • John Dauns
  • Yiqiang Zhou
Part of the Trends in Mathematics book series (TM)

Abstract

This paper is about QI-modules M and the full subcategory σ[M] of Mod-R subgenerated by M. A ring R is a right QI-ring if every quasi-injective right R-module is injective. The module M is QI if all quasi-injective modules in σ[M] are M-injective. Three classes of rings are presented for each of which the QI-modules are precisely the semisimple modules. The QI-modules M are characterized in terms of properties of some lattices of classes of modules in σ[M].

Keywords

Direct Summand Module Class Subdirect Product Natural Class Boolean Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. Beachy and W.D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), 1–13.MathSciNetzbMATHGoogle Scholar
  2. [2]
    J. Dauns, Unsaturated classes of modules, Abelian groups and modules (Colorado Springs, CO, 1995), pp. 211–225, Lecture Notes in Pure and Appl. Math., 182, Dekker, New York, 1996.Google Scholar
  3. [3]
    J. Dauns and Y. Zhou, Classes of Modules, Pure and Applied Mathematics (Boca Raton), 281, Chapman & Hall/CRC, Boca Raton, FL, 2006.Google Scholar
  4. [4]
    L. Fuchs, On quasi-injective modules, Annali dela Scuola Norm. Sup. Pisa 23 (1969), 541–546.zbMATHGoogle Scholar
  5. [5]
    J.L. García Hernández and J.L. Gómez Pardo, V-rings relative to Gabriel topologies, Comm. Alg. 13 (1985), 58–83.CrossRefGoogle Scholar
  6. [6]
    R. Gordon and J.C. Robson, Krull Dimension, Mem. Amer. Math. Soc. 133, 1973.Google Scholar
  7. [7]
    R. Gordon and J.C. Robson, The Gabriel dimension of a module, J. Algebra 29 (1974), 459–473.CrossRefMathSciNetzbMATHGoogle Scholar
  8. [8]
    S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambridge University Press, New York, 1990.Google Scholar
  9. [9]
    S. Page and Y. Zhou, On direct sums of injective modules and chain conditions, Canad. J. Math. 46 (1994), 634–647.MathSciNetzbMATHGoogle Scholar
  10. [10]
    F. Raggi, J.R. Montes and R. Wisbauer, The lattice structure of hereditary pretorsion classes, Comm. Algebra 29 (2001), 131–140.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    B. de la Rosa, and G. Viljoen, A note on quasi-injective modules, Comm. Alg., 15(6) (1987), 1279–1286.CrossRefzbMATHGoogle Scholar
  12. [12]
    M.L. Teply, On the idempotence and stability of kernel functors, Glasgow Math. J. 37 (1995), 37–43.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    R. Wisbauer, Generalized co-semisimple modules, Comm. Algebra 18(12) (1990), 4235–4253.MathSciNetzbMATHGoogle Scholar
  14. [14]
    R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, 1991.Google Scholar
  15. [15]
    R. Wisbauer, On module classes closed under extensions, in: Rings and Radicals, pp. 73–97, Pitman Res. Notes Math. Ser., 346, Longman, Harlow, 1996.Google Scholar
  16. [16]
    R. Wisbauer, Cotilting objects and dualities, Representations of algebras (Sao Paulo, 1999), 215–233, Lecture Notes in Pure and Appl. Math., 224, Dekker, New York, 2002.Google Scholar
  17. [17]
    Y. Zhou, Direct sums of M-injective modules and module classes, Comm. Algebra, 23(3) 1995, 927–940.CrossRefMathSciNetzbMATHGoogle Scholar
  18. [18]
    Y. Zhou, The lattice of pre-natural classes of modules, J. Pure Appl. Algebra 140(2) (1999), 191–207.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • John Dauns
    • 1
  • Yiqiang Zhou
    • 2
  1. 1.Tulane UniversityNew OrleansUSA
  2. 2.Memorial University of NewfoundlandSt.John’sCanada

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