When Does the Complement of the Annihilating-Ideal Graph of A Commutative Ring Admit A Cut Vertex?

The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. let r be a ring. let a(r) denote the set of all annihilating ideals of r and let a(r)* = a(r) [(0)]. the annihilating-ideal graph of r,denoted by ag(r) is an undirected simple graph whose vertex set is a(r)* and distinct, vertices i, j are joined by an edge in this graph if and only if ij = (0). the aim of this article is to classify rings r such that (ag(r))^c ( that is, the complement of ag(r)) is connected and admits a cut vertex.