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A Survey of Rings Satisfying Annihilator or Extending Conditions on Projection Invariant Ideals

  • Gary F. BirkenmeierEmail author
  • Yeliz Kara
  • Adnan Tercan
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 277)

Abstract

In this paper, we survey results involving the projection invariant condition on one-sided ideals of rings. We focus on rings satisfying the right projection invariant extending condition (denoted right \(\pi \)-extending) or the projection invariant Baer condition (denoted \(\pi \)-Baer). Examples are provided to illustrate and delimit the results.

Keywords

Baer ring Quasi-Baer ring Extending module Extending ring FI-extending Projection invariant Ring extension 

2010 AMS Subject Classification:

16D10 16S36 16S50 16N60 16S70 16S75 16W60 

Notes

Acknowledgements

We wish to thank the referee and the editor for their efforts.

References

  1. 1.
    Abdioğlu, C., Koşan, M.T., Şahinkaya, S.: On modules for which all submodules are projection invariant and the lifting condition. Southeast Asian Bull. Math. 5, 807–818 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Armendariz, E.P.: A note on extensions of Baer and pp-rings. J. Austral. Math. Soc. 18, 470–473 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berberian, S.K.: Baer *-Rings. Grundlehren der mathematischen Wissenschaften, vol. 195. Springer, Berlin (1972)Google Scholar
  4. 4.
    Birkenmeier, G.F., Müller, B.J., Rizvi, S.T.: Modules in which every fully invariant submodule is essential in a direct summand. Commun. Algebra 30, 1395–1415 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Birkenmeier, G.F., Tercan, A.: When some complement of a submodule is a direct summand. Commun. Algebra 35, 597–611 (2007)CrossRefGoogle Scholar
  6. 6.
    Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Extensions of Rings and Modules. Birkhäuser, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Birkenmeier, G.F., Lennon, M.J.: Extending sets of idempotents to ring extensions. Commun. Algebra 42, 5134–5151 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Birkenmeier, G.F., Lennon, M.J.: Dense intrinsic extensions. Houston J. Math. 40, 21–42 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Birkenmeier, G.F., Tercan, A., Yücel, C.C.: The extending condition relative to sets of submodules. Commun. Algebra 42, 764–778 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Birkenmeier, G.F., Tercan, A., Yücel, C.C.: Projection invariant extending rings. J. Algebra Appl. 115, 1650121 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Birkenmeier, G.F., Kara, Y., Tercan, A.: \(\pi \)-Baer rings. J. Algebra Appl. 17(2), 1850029 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chatters, A.W., Khuri, S.M.: Endomorphism rings of modules over nonsingular CS-rings. J. London Math. Soc. 21, 434–444 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Clark, W.E.: Twisted matrix units semigroup algebras. Duke Math. J. 34, 417–423 (1967)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dung, N.V., Van Huynh, D., Smith, P.F., Wisbauer, R.: Extending Modules. Longman, England (1994)zbMATHGoogle Scholar
  15. 15.
    Faith, C., Utumi, Y.: Intrinsic extensions of rings. Pacific J. Math. 14, 505–512 (1964)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fuchs, L.: Infinite Abelian Groups I. Academic Press, New York (1970)zbMATHGoogle Scholar
  17. 17.
    Goodearl, K.R.: Von Neumann Regular Rings. Krieger, Malabar (1991)zbMATHGoogle Scholar
  18. 18.
    Kaplansky, I.: Rings of Operators. Benjamin, New York (1968)zbMATHGoogle Scholar
  19. 19.
    Lam, T.Y.: Lectures on Modules and Rings. Springer, Berlin (1999)CrossRefGoogle Scholar
  20. 20.
    Megibben, C.K.: Projection invariant subgroups of Abelian groups. Tamkang J. Math. 2, 177–182 (1977)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mohamed, S.H., Muller, B.J.: Continuous and Discrete Modules. Cambridge Univ. Press, Cambridge (1990)CrossRefGoogle Scholar
  22. 22.
    Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math. 37, 116–229 (1936)MathSciNetCrossRefGoogle Scholar
  23. 23.
    von Neumann, J.: Continuous geometry. Proc. Math. Acad. Sci. 22, 92–100 (1936)CrossRefGoogle Scholar
  24. 24.
    Pollingher, A., Zaks, A.: On Baer and quasi-Baer rings. Duke Math. J. 37, 127–138 (1970)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rickart, C.E.: Banach Algebras with an adjoint operation. Ann. Math. 47, 528–550 (1946)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sikander, F., Mehdi, A., Naji, S.A.R.K.: On projection invariant submodules of QTAG-modules. J. Egyptian Math. Soc. 24, 156–159 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tercan, A., Yücel, C.C.: Module Theory, Extending Modules and Generalizations. Birkhäuser, Basel (2016)CrossRefGoogle Scholar
  28. 28.
    Utumi, Y.: On continuous rings and self injective rings. Trans. Amer. Math. Soc. 118, 158–173 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gary F. Birkenmeier
    • 1
    Email author
  • Yeliz Kara
    • 2
  • Adnan Tercan
    • 3
  1. 1.Department of MathematicsUniversity of Louisiana at LafayetteLafayetteUSA
  2. 2.Department of MathematicsBursa Uludağ UniversityBursaTurkey
  3. 3.Department of MathematicsHacettepe UniversityAnkaraTurkey

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