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

Suppose that RR is a commutative ring with identity. Let A(R)A(R) be the set of all ideals of RR with non-zero annihilators. The annihilating-ideal graph of RR is defined as the graph AG(R)AG(R) with the vertex set A(R)∗=A(R)∖{(0)}A(R)∗=A(R)∖{(0)} and two distinct vertices II and JJ are adjacent if and only if IJ=(0)IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring RR with more than two minimal prime ideals, girth(AG(R))=3girth(AG(R))=3. Here, we prove that for every (not necessarily reduced) ring RR, ω(AG(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 AG(R)AG(R) is bipartite if and only if AG(R)AG(R) is triangle-free.