When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

Abstract EN

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 Z/3Z×Z/3Z as rings.

Moreover, if R is not isomorphic to Z/3Z×Z/3Z 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.