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

Abstract

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 AG(R) with the vertex set A(R)∗=A(R)∖{(0)} and two distinct vertices I and J are adjacent if and only if IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R))=3. Here, we prove that for every (not necessarily reduced) ring R, 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) is bipartite if and only if AG(R) is triangle-free.