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Baer and Quasi-Baer Modules over Some Classes of Rings

  • Haily, Abdelfattah ;
  • Rahnaou, Hamid
  • Received : 2011.01.27
  • Accepted : 2011.08.21
  • Published : 2011.11.23

Abstract

We study Baer and quasi-Baer modules over some classes of rings. We also introduce a new class of modules called AI-modules, in which the kernel of every nonzero endomorphism is contained in a proper direct summand. The main results obtained here are: (1) A module is Baer iff it is an AI-module and has SSIP. (2) For a perfect ring R, the direct sum of Baer modules is Baer iff R is primary decomposable. (3) Every injective R-module is quasi-Baer iff R is a QI-ring.

Keywords

Endomorphism;Idempotent;Annihilator;Baer module;$\mathcal{K}$-nonsingular module

References

  1. M. Alkan and A. Harmanci, On Summand Sum and Summand Intersection Property of Modules, Turk. J. Math., 26(2002), 131-147.
  2. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second edition, GTM 13, Springer-Verlag, New-York, 1992.
  3. S. K. Berberian, Baer *-Rings, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
  4. W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J., 34(1967), 417-424. https://doi.org/10.1215/S0012-7094-67-03446-1
  5. C. Faith, Algebra II, Ring theory, Springer-Verlag, 1981.
  6. A. Haily and M. Alaoui, Perfect Rings for which the converse of Schur's lemma holds, Publ. Mat., 45(2001), 219-222. https://doi.org/10.5565/PUBLMAT_45101_10
  7. A. Haily, H. Rahnaoui, Some external characterizations of SV-rings and hereditary rings, Int. J. Math. Math. Sci., 2007, Art. ID 84840, 6 pp https://doi.org/10.1016/B978-012373947-6/00488-8
  8. I. Kaplansky, Rings of operators, New York, Benjamin, 1968.
  9. S. T. Rizvi, C. S. Roman, Baer and quasi-Baer modules, Comm. Algebra, 32(2004), 103-123. https://doi.org/10.1081/AGB-120027854
  10. S. T. Rizvi, C. S. Roman, On K-nonsingular modules and applications, Comm. Algebra, 35(2007), 2960-2982 https://doi.org/10.1080/00927870701404374
  11. S. T. Rizvi, C. S. Roman, On direct sums of Baer modules, J. Algebra 321(2009), 682-696. https://doi.org/10.1016/j.jalgebra.2008.10.002
  12. T. Y. Lam Lectures on modules and rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999.
  13. G. V. Wilson. Modules with the summand intersection property, Comm. Alg., 14(1)(1986). 21-38 https://doi.org/10.1080/00927878608823297
  14. R. Wisbauer, Foundations of module and ring theory, Gordon and Beach Science Publishers, 1991.

Cited by

  1. On weak Rickart modules vol.16, pp.09, 2017, https://doi.org/10.1142/S0219498817501651
  2. Direct sums of quasi-Baer modules vol.456, 2016, https://doi.org/10.1016/j.jalgebra.2016.01.039
  3. Weak Rickart and dual weak Rickart objects in abelian categories vol.46, pp.7, 2018, https://doi.org/10.1080/00927872.2017.1404079