Skip to main content
SpringerLink
Log in

A graph over the commutative rings

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

  • 1 Altmetric

Abstract

In this paper, we introduce a new graph over a commutative ring R relative to an R-module M. We study the relationship between the algebraic properties of R and M and their associated graph, say G(R, M). More precisely, we consider the relationship between the completeness of subgraphs of the G(R, M) and algebraic properties of R and M. In addition, we study the bipartite and complete bipartite subgraphs of the associated graph. Finally, we completely characterize the diameter of the certain subgraph of G(R, M).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akbari, S., Maimani, H.R., Yassemi, S.: When a zero-divisor graph is planar or a complete \(r\)-partite graph. J. Algebra 270, 169–180 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbari, S., Mohammadian, A.: Zero-divisor graphs of non-commutative rings. J. Algebra 296, 462–479 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, D., Badawi, A.: On the zero-divisor graph of a ring. Commun. Algebra 36, 3073–3092 (2008)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, D., Mulay, S.: On the diameter and girth of a zero-divisor graph. J. Pure Appl. Algebra 210(2), 543–550 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Anderson, F.W., Fuller, K.R.: Rings and categories of modules. No. 13. In: Graduate Texts in Mathematics. Springer, New York (1992)

  7. Axtell, M., Coykendall, J., Stickles, J.: Zero-divisor graphs of polynomial and power series over commutative rings. Commun. Algebra 33, 2043–2050 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Azarpanah, F., Motamedi, M.: Zero-divisor graph of C(X). Acta Math. Hung. 108, 25–36 (2005)

    Article  MATH  Google Scholar 

  9. Beck, I.: Coloring of commutative rings. J. Algebra 116(1), 208–226 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brodmann, M.P., Sharp, R.Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge University Press, Cambridge (1998)

  11. LaGrange, J.D.: Complemented zero-divisor graphs and Boolean rings. J. Algebra 315, 600–611 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lu, C.P.: Prime submodules of modules. Comment. Math. Univ. St. Pauli 33(1), 61–69 (1984)

    MathSciNet  MATH  Google Scholar 

  13. Lu, C.P.: A module whose prime spectrum has the surjective natural map. Houst. J. Math. 33(1), 125–143 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Maimani, H., Salimi, M., Sattari, A., Yassemi, S.: Comaximal graph of commutative rings. J. Algebra 319, 1801–1808 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sharma, P., Bhatwadekar, S.: A note on graphical representation of rings. J. Algebra 176, 124–127 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, H.: Graphs associated to co-maximal ideals of commutative rings. J. Algebra 320, 2917–2933 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach, New York (1991)

  18. Zöschinger, H.: Koatomare moduln. Math. Z. 170, 221–232 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors specially thank the referee for the helpful suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Fouman and Shaft Branch, Islamic Azad University, Guilan, Iran

    Leila Hamedi-Mobarra

  2. Department of Basic Sciences, Arak University of Technology, P.O. Box 38135-1177, Arak, Markazi, Iran

    Dawood Hassanzadeh-lelekaami

  3. Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Iran

    Hajar Roshan-Shekalgourabi

Authors
  1. Leila Hamedi-Mobarra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dawood Hassanzadeh-lelekaami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hajar Roshan-Shekalgourabi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leila Hamedi-Mobarra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hamedi-Mobarra, L., Hassanzadeh-lelekaami, D. & Roshan-Shekalgourabi , H. A graph over the commutative rings. Bol. Soc. Mat. Mex. 23, 611–621 (2017). https://doi.org/10.1007/s40590-016-0132-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-016-0132-8

Keywords

Mathematics Subject Classification

Search

Navigation