Skip to main content
Log in

Abstract

Let R be a commutative ring with nonzero identity and Z(R) be its set of zero-divisors. The new extension of the zero-divisor graph \(\widetilde{\Gamma }(R)\) with vertices \(Z(R)^{\star }\), and two distinct vertices x and y are adjacent if and only if \(xy=0\) or \(x+y\in Z(R)\). In this article, we study, in the general case, the graph \(\widetilde{\Gamma }(R)\). For any commutative ring R, we provide sufficient and necessary conditions for \(\widetilde{\Gamma }(R)\) and \(\widetilde{\Gamma }(R[x_1,\dots , x_n])\) to be complete. At last, we present some other properties of this new extension of the zero-divisor graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Boston (1969)

    MATH  Google Scholar 

  • Anderson, D.F., Badawi, A.: The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)

    Article  MathSciNet  Google Scholar 

  • Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)

    Article  MathSciNet  Google Scholar 

  • Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)

    Article  MathSciNet  Google Scholar 

  • Cherrabi, A., Essannouni, H., Jabbouri, E., Ouadfel, A.: On a new extension of the zero-divisor graph. To appear in Algebra Colloquium. 2020 (arXiv:1806.11442)

  • Diestel, R.: Graph Theory. Springer-Verlag, New York (1997)

    MATH  Google Scholar 

  • Hazewinkel, M., Gubareni, N., Kirichenko, V.V.: Algebras, Rings and Modules, vol. 1, Mathematics and its applications, Kluwer Academic Publishers (2005)

  • Huckaba, J.: Commutative Rings with Zero Divisors. Dekker, New York (1988)

    MATH  Google Scholar 

  • Lucas, T.G.: The diameter of zero divisor graph. J. Algebra 301, 174–193 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for careful reading and helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Cherrabi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cherrabi, A., Essannouni, H., Jabbouri, E. et al. On a new extension of the zero-divisor graph (II). Beitr Algebra Geom 62, 945–953 (2021). https://doi.org/10.1007/s13366-020-00559-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-020-00559-8

Keywords

Mathematics Subject Classification

Navigation