• 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)
  • Received : 2014.06.24
  • Published : 2015.03.01


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.


annihilator ideal graph;diameter;Clique number


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