Rings whose associated extended zero-divisor graphs are complemented

Let \(R\) be a commutative ring with identity \(1\neq 0\). In this paper, we continue the study started in [10] concerning when the extended zero-divisor graph of \(R\), \(\overline{\Gamma}(R)\), is complemented. We also study when \(\overline{\Gamma}(R)\) is uniquely complemented. We give a complete characterization of when \(\overline{\Gamma}(R)\) of a finite ring is complemented. Various examples are given using the direct product of rings and idealizations of modules.