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Siberian Mathematical Journal

, Volume 60, Issue 3, pp 490–496 | Cite as

Decompositions of Dual Automorphism Invariant Modules over Semiperfect Rings

  • Y. KuratomiEmail author
Article
  • 19 Downloads

Abstract

A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

Keywords

dual automorphism invariant module pseudoprojective module dual square free module finite internal exchange property (semi)perfect ring 

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References

  1. 1.
    Clark J., Lomp C., Vanaja N., and Wisbauer R., Lifting Modules. Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkhauser, Boston (2006).zbMATHGoogle Scholar
  2. 2.
    Mohamed S. H. and Müller B. J., Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge (1990) (Lond. Math. Soc. Lect. Note Ser.; V. 147).CrossRefzbMATHGoogle Scholar
  3. 3.
    Singh S. and Srivastava A. K., “Dual automorphism-invariant modules,” J. Algebra, vol. 371, 262–275 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guil Asensio P. A., Keskin Tütüncü D., Kaleboḡaz B., and Srivastava A. K., “Modules which are coinvariant under automorphisms of their projective covers,” J. Algebra, vol. 466, 147–152 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Abyzov A. N., Quynh T. C., and Tai D. D., “Dual automorphism-invariant modules over perfect rings,” Sib. Math. J., vol. 58, no. 5, 743–751 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guil Asensio P. A., Keskin Tuütuüncuü D., and Srivastava A. K., “Modules invariant under automorphisms of their covers and envelopes,” Israel J. Math., vol. 206, no. 1, 457–482 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Keskin Tütüncü D., “When automorphism-coinvariant modules are quasi-projective,” Comm. Algebra, vol. 45, 688–693 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lee T.-K. and Zhou Y., “Modules which are invariant under automorphisms of their injective hulls,” J. Algebra Appl., vol. 12, no. 2, 1250159 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Tuganbaev A. A., “Automorphism-invariant modules,” J. Math. Sci., vol. 206, no. 6, 694–698 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Baba Y. and Oshiro K., Classical Artinian Rings and Related Topics, World Sci. Publ. Co. Pte. Ltd, Hackensack (NJ) (2009).CrossRefzbMATHGoogle Scholar
  11. 11.
    Wisbauer R., Foundations of Module and Ring Theory, Gordon and Breach, Reading (1991).zbMATHGoogle Scholar
  12. 12.
    Keskin Tuütuüncuü D., Kikumasa I., Kuratomi Y., and Shibata Y., “On dual of square free modules,” Comm. Algebra, vol. 46, 3365–3376 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kikumasa I. and Kuratomi Y., “On H-supplemented modules over a right perfect ring,” Comm. Algebra, vol. 46, 2063–2072 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuratomi Y., “H-supplemented modules and generalizations of quasi-discrete modules,” Comm. Algebra, vol. 44, 2747–2759 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuratomi Y., “On direct sums of lifting modules and internal exchange property,” Comm. Algebra, vol. 33, 1795–1804 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceYamaguchi UniversityYoshida, YamaguchiJapan

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