Ideal Graphs Supported By Given Ideals of Commutative Rings

Authors

  • Fryad H. Abdulqadr Mathematics Department, College of Education, Salahaddin University, Erbil, Kurdistan Region, Iraq. Author

DOI:

https://doi.org/10.17656/jzs.10786

Keywords:

Ideal graph supported by given ideals of commutative rings, Connected graphs, Clique and chromatic number

Abstract

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

References

Akbari, S. and Mohamadian, A., "On the Zero-Divisor Graph of a Commutative Ring", Journal of Algebra. Vol. 274, No. 2, pp.847-855. (2004). DOI: https://doi.org/10.1016/S0021-8693(03)00435-6

Andersn, D. F. and Livingston, P. S., "The zero divisor graph of a commutative ring", Journal of Algebra. Vol. 217, pp. 434-447. (1999). DOI: https://doi.org/10.1006/jabr.1998.7840

Beck, I., "Coloring of commutative rings", Journal of Algebra. 116, pp. 208–226. (1988). DOI: https://doi.org/10.1016/0021-8693(88)90202-5

Behboodi, M. and Rakeei, Z., "The Annihilating-Ideal Graph of Commutative Rings I", Journal of Algebra and Appl., Vol.10, No.4, pp. 727-739. (2011). DOI: https://doi.org/10.1142/S0219498811004896

David, S. and Richard, M., "Abstract Algebra", Prentice-Hall Inc. U. S. A. (1991).

Gary, C. and Linda, L., "Graphs and Digraphs", 2nd ed., Wadsworth and Brooks/Cole, California. (1986).

Gupta, R, Sen, S. M. K. and Ghosh, S., " A variation of zero-divisor graphs", Discuss. Math. Gen. Algebra Appl. Vol. 35, No. 2, pp. 159–176. (2015). DOI: https://doi.org/10.7151/dmgaa.1238

Mahadevi, P. and Babujee, J., " On Special Structures of Zero-Divisor Graphs", International Journal of Pure and Applied Mathematics. Vol. 119, No. 13, pp. 281-287. (2018)

Patil, A., Waphare, B. N. and Joshi, V., "Perfect zero-divisor graphs", Discrete Math. Vol. 340, No. 4, pp. 740–745. (2017) DOI: https://doi.org/10.1016/j.disc.2016.11.027

Shuker, N. H., Mohammad, H. Q. and Luma, A. K., "Hosoya and Wiener Index of Zero-Divisor Graph of ,"Zanco Journal of Pure and Applied Sciences. Vol. 31, No. 2, pp. 45-52. (2019). DOI: https://doi.org/10.21271/ZJPAS.31.s2.6

Published

2020-06-20

How to Cite

Ideal Graphs Supported By Given Ideals of Commutative Rings. (2020). Journal of Zankoy Sulaimani - Part A, 22(1), 217-222. https://doi.org/10.17656/jzs.10786

Most read articles by the same author(s)

1 2 3 4 5 6 7 8 9 10 > >>