On the clique number of the complement of the annihilating ideal graph of a commutative ring

Abstract

The rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. Let \(R\) be such a ring. If \(Z(R)\), the set of zero-divisors of \(R\), is not an ideal of \(R\), then it is shown in this article that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique if and only if its clique number is finite. Moreover, this article classifies (up to isomorphism of rings) rings \(R\) such that \(Z(R)\) is not an ideal and for which the complement of its annihilating ideal graph does not admit any infinite clique. Furthermore, the clique number of the complement of the annihilating ideal graph of \(R\) is determined under the assumption that \(R\) is reduced with the property that the complement of its annihilating ideal graph does not contain any infinite clique. In addition, this article also considers rings \(R\) such that \(Z(R)\) is an ideal of \(R\) and some necessary conditions are determined in order that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique. Moreover, for a chained ring \(R\), it is shown that the complement of the annihilating ideal graph of \(R\) does not contain any infinite clique if and only if its clique number is finite. The clique number of the complement of the annihilating ideal graph of a chained ring \(R\) is determined under the hypothesis that it does not contain any infinite clique.