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On two extensions of the classical zero-divisor graph

  • Malik Bataineh
  • Driss BennisEmail author
  • Jilali Mikram
  • Fouad Taraza
Original Article
  • 22 Downloads

Abstract

The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.

Keywords

Zero-divisor graphs Extended zero-divisor graphs Annihilator graphs 

Mathematics Subject Classification

13A99 13B99 05C25 

Notes

Acknowledgements

The authors would like to thank the referees for careful reading of the manuscript and helpful comments.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. 1.
    Anderson, D.F.: On the diameter and girth of a zero-divisor graph, II. Houst. J. Math. 34, 361–371 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, D.F., Badawi, A.: Divisibility conditions in commutative rings with zero-divisors. Commun. Algebra 30, 4031–4047 (2002)CrossRefGoogle Scholar
  3. 3.
    Anderson, D.F., Badawi, A.: On the zero-divisor graph of a ring. Commun. Algebra 36, 3073–3092 (2008)CrossRefGoogle Scholar
  4. 4.
    Anderson, D.F., Badawi, A.: The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Anderson, D.F., Badawi, A.: The generalized total graph of a commutative ring. J. Algebra Appl. 12, 1250212–1250230 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Anderson, D.F., Mulay, S.B.: On the diameter and girth of a zero-divisor graph. J. Pure Appl. Algebra 210, 543–550 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Anderson, D.D., Naseer, M.: Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Anderson, D.D., Winders, M.: Idealization of a module. J. Commut. Algebra 1, 3–56 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Anderson, D.F., Levy, R., Shapiro, J.: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180, 221–241 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Anderson, D.F., Axtell, M., Stickles, J.: Zero-divisor graphs in commutative rings. In: Fontana, M., Kabbaj, S.-E., Olberding, B., Swanson, I. (eds.) Commutative Algebra, Noetherian and Non-Noetherian Perspectives, pp. 23–45. Springer, New York (2010)Google Scholar
  12. 12.
    Axtell, M., Stickles, J.: Zero-divisor graphs of idealizations. J. Pure Appl. Algebra 204, 235–243 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Axtell, M., Coykendall, J., Stickles, J.: Zero-divisor graphs of polynomial and power series over commutative rings. Commun. Algebra 33, 2043–2050 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Badawi, A.: On the annihilator graph of a commutative ring. Commun. Algebra 42, 108–121 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bennis, D., Mikram, J., Taraza, F.: On the extended zero-divisor graph of commutative rings. Turk. J. Math. 40, 376–388 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bennis, D., Mikram, J., Taraza, F.: Extended zero-divisor graph of idealizations. Commun. Korean Math. Soc. 32, 7–17 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bollabos, B.: Modern Graph Theory. Springer, New York (1998)CrossRefGoogle Scholar
  19. 19.
    Huckaba, J.A.: Commutative Rings with Zero Divisors. Marcel Dekker, New York (1988)zbMATHGoogle Scholar
  20. 20.
    Levy, R., Shapiro, J.: The zero-divisor graph of von Neumann regular rings. Commun. Algebra 30, 745–750 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lucas, T.G.: The diameter of a zero divisor graph. J. Algebra 301, 3533–3558 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Maimani, H.R., Salimi, M., Sattari, A., Yassemi, S.: Comaximal graph of commutative rings. J. Algebra 319, 1801–1808 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan
  2. 2.Centre de Recherche de Mathématiques et Applications de Rabat (CeReMAR), Faculty of SciencesMohammed V University in RabatRabatMorocco

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