When is the complement of the zero-divisor graph of a commutative ring complemented? (Q448546)

Summary: Let \(R\) be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of \(R\) has at least one edge. Under the above assumptions on \(R\), it is shown in this paper that the complement of the zero-divisor graph of \(R\) is complemented if and only if \(R\) is isomorphic to \(\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\) as rings. Moreover, if \(R\) is not isomorphic to \(\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\) as rings, then, it is shown that in the complement of the zero-divisor graph of \(R\), either no vertex admits a complement or there are exactly two vertices which admit a complement.