On the strongly annihilating-ideal graph of a commutative ring

Abstract

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. It is proved that [Formula: see text] is connected with diameter at most two and with girth at most four, if it contains a cycle. Moreover, we characterize all rings whose strongly annihilating-ideal graphs are complete or star. Furthermore, we study the affinity between strongly annihilating-ideal graph and annihilating-ideal graph (a well-known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with a commutative ring.