When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?

Abstract

Let R be a commutative ring with identity. Let A(R) denote the collection of all annihilating ideals of R (that is, A(R) is the collection of all ideals I of R which admits a nonzero annihilator in R). Let AG(R) denote the annihilating ideal graph of R. In this article, necessary and sufficient conditions are determined in order that AG(R) is complemented under the assumption that R is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R such that AG(R) is complemented under the assumption that A(R) contains at least two nonzero ideals.