## Taiwanese Journal of Mathematics

### DOMINATION IN THE ZERO-DIVISOR GRAPH OF AN IDEAL OF A NEAR-RING

#### Abstract

Let $N$ be a near-ring. In this paper, we associate a graph corresponding to the $3$-prime radical $\mathcal{I}$ of $N$, denoted by $\Gamma_\mathcal{I}(N)$. Further we obtain certain topological properties of $Spec(N)$, the spectrum of $3$-prime ideals of $N$ and graph theoretic properties of $\Gamma_\mathcal{I}(N)$. Using these properties, we discuss dominating sets and connected dominating sets of $\Gamma_\mathcal{I}(N)$.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 5 (2013), 1613-1625.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706228

Digital Object Identifier
doi:10.11650/tjm.17.2013.2739

Mathematical Reviews number (MathSciNet)
MR3106033

Zentralblatt MATH identifier
1301.16053

#### Citation

Tamizh Chelvam, T.; Nithya, S. DOMINATION IN THE ZERO-DIVISOR GRAPH OF AN IDEAL OF A NEAR-RING. Taiwanese J. Math. 17 (2013), no. 5, 1613--1625. doi:10.11650/tjm.17.2013.2739. https://projecteuclid.org/euclid.twjm/1499706228

#### References

• G. Alan Cannon, Kent M. Neuerburg and Shane P. Redmond, Zero-divisor graphs of Near-rings and Semigroups, Near-rings and Near-fields:Proceedings of the Conference on Near-rings and Near-fields II, 2005, pp. 189-200, doi: 10.1007/1-4020-3391-5_8.
• D. F. Anderson and P. S. Livingston, The zero divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
• I. Beck, Coloring of Commutative rings, J. Algebra, 116 (1988), 208-226.
• N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1973.
• G. F. Birkenmeier, H. E. Heatherly and E. K Lee, Completely prime ideals and radicals in near-rings, Proc. Fredericton Conference on Near-rings and Near-fields, Kluwer Acad. Publ., Dordrecht, 1995, pp. 63-67.
• G. Chartrand and P. Zhang, Introduction to Graph Theory, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
• P. Dheena and B. Elavarasan, An ideal-based zero-divisor graph of $2$-primal near-rings, Bull. Korean Math. Soc., 46 (2009), 1051-1060.
• R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
• J. R. Munkres, Topology, Prentice-Hall of India, New Delhi, 2005.
• G. Pilz, Near-rings, North Holland, Amsterdam, 1983.
• S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31 (2003), 4425-4443.
• K. Samei, The zero-divisor graph of a reduced ring, J. Pure Appl. Algebra, 209 (2007), 813-821.
• T. Tamizh Chelvam and S. Nithya, Zero-divisor graph of an ideal of a near-ring, Discrete Math. Algorithms Appl., to appear.
• S. Veldsman, On equiprime near-rings, Comm. Algebra 20(9) (1992), 2569-2587.