On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal

Authors

  • Priyanka Pratim Baruah Department of Mathematics, Girijananda Chowdhury Institute of Management and Technology, Guwahati 781017
  • Kuntala Patra Department of Mathematics, Gauhati University, Guwahati 781014

DOI:

https://doi.org/10.26713/cma.v8i3.355

Keywords:

Commutative ring, Ideal, Generalized zero-divisor graph, Diameter, Girth.

Abstract

In this paper, we generalize the notion of the ideal-based zero-divisor graph of a commutative ring. Let \(R\) be a commutative ring and let \(I\) be an ideal of \(R\). Here, we define a generalized zero-divisor graph of \(R\) with respect to \(I\) and denote this graph by \(\Gamma_I^G(R)\). We show that \(\Gamma_I^G(R)\) is connected with diameter at most three. If \(\Gamma_I^G(R)\) has a cycle, we show that the girth of \(\Gamma_I^G(R)\) is at most four. Also, we investigate the existence of cut vertices of \(\Gamma_I^G(R)\). Moreover,we examine certain situations when \(\Gamma_I^G(R)\) is a complete bipartite graph.

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References

S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), 847– 855.

D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500–514.

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

D. F. Anderson, A. Frazier, A. Laave and P. S. Livingston, The zero-divisor graph of a commutative ring II, Lecture notes in pure and appl. math., 220 (2001), New York/Basel:

Marcel Dekker, 61 – 72.

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

P. Dheena and B. Elavarasan, An ideal-based zero-divisor graph of 2-primal near-rings, Bull. Commun. Korean Math. Soc. 46 (2009), No. 6, 1051–1060.

P. Dheena and B. Elavarasan, A generalized ideal-based zero-divisor graph of a near-ring, Commun. Korean Math. Soc. 24 (2009), No. 2, 161–169.

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

N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, (Asia) Pte Ltd (1999).

H. Maimani, M. R. Pouranki and S. Yassemi, Zero-divisor graph with respect to an ideal, Communications in Algebra, Vol. 34 (2006) no. 3, 923 – 929.

S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31 (2003), No. 9, 4425– 4443.

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Published

30-12-2017
CITATION

How to Cite

Baruah, P. P., & Patra, K. (2017). On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal. Communications in Mathematics and Applications, 8(3), 333–343. https://doi.org/10.26713/cma.v8i3.355

Issue

Vol. 8 No. 3 (2017)

Section

Research Article

License

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