On the coloring of the annihilating-ideal graph of a commutative ring

Suppose that R is a commutative ring with identity. Let A ( R ) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph A G ( R ) with the vertex set A ( R ) ∗ = A ( R ) ∖ { ( 0 ) } and two distinct vertices I and J are adjacent if and only if I J = ( 0 ) . In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, g i r t h ( A G ( R ) ) = 3 . Here, we prove that for every (not necessarily reduced) ring R , ω ( A G ( R ) ) ≥ | Min ( R ) | , which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate commutative rings whose annihilating-ideal graphs are bipartite. It is proved that A G ( R ) is bipartite if and only if A G ( R ) is triangle-free.