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http://dx.doi.org/10.4134/JKMS.2015.52.2.417

THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING  

Alibemani, Abolfazl (Faculty of Mathematics K. N. Toosi University of Technology)
Bakhtyiari, Moharram (Department of Mathematics College of Basic Sciences Karaj Branch, Islamic Azad University)
Nikandish, Reza (Department of Mathematics Jundi-Shapur University of Technology)
Nikmehr, Mohammad Javad (Faculty of Mathematics K. N. Toosi University of Technology)
Publication Information
Journal of the Korean Mathematical Society / v.52, no.2, 2015 , pp. 417-429 More about this Journal
Abstract
Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if $I{\cap}Ann(J){\neq}\{0\}$ or $J{\cap}Ann(I){\neq}\{0\}$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if ${\Gamma}_{Ann}(R)$ is triangle free, then R is Gorenstein.
Keywords
annihilator ideal graph; diameter; Clique number;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 S. Akbari, R. Nikandish, and M. J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013), no. 4, 1250200, 13 pp.   DOI
2 D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447.   DOI   ScienceOn
3 I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226.   DOI
4 W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1997.
5 J. Chen, N. Ding, and M. F. Yousif, On Noetherian rings with essential socle, J. Aust. Math. Soc. 76 (2004), no. 1, 39-49.   DOI
6 S. Ebrahimi Atani, S. Dolati Pish Hesari, and M. Khoramdel, Total graph of a commu- tative semiring with respect to identity-summand elements, J. Korean Math. Soc. 51 (2014), no. 3, 593-607.   DOI   ScienceOn
7 F. Heydari and M. J. Nikmehr, The unit graph of a left Artinian ring, Acta Math. Hungar. 139 (2013), no. 1-2, 134-146.   DOI
8 J. A. Huckaba, Commutative Rings with Zero-Divisors, Marcel Dekker, Inc., New York, 1988.
9 T. Y. Lam, A First Course in Non-commutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, Berlin/Heidelberg, New York, 1991.
10 R. Nikandish and M. J. Nikmehr, The intersection graph of ideals of $Z_{n}$ is weakly perfect, Util. Math., to appear.
11 R. Y. Sharp, Steps in Commutative Algebra, Second Edition, London Mathematical Society Student Texts 51, Cambridge University Press, Cambridge, 2000.
12 D. B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Upper Saddle River, 2001.
13 R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, 1991.