# 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)
• Published : 2015.03.01
• 72 10

#### 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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#### Cited by

1. On the essential graph of a commutative ring vol.16, pp.07, 2017, https://doi.org/10.1142/S0219498817501328
2. Some results on the strongly annihilating-ideal graph of a commutative ring 2017, https://doi.org/10.1007/s40590-017-0179-1
3. On the strongly annihilating-ideal graph of a commutative ring vol.09, pp.02, 2017, https://doi.org/10.1142/S1793830917500288