Some Properties of the Complement of the Zero-Divisor Graph of a Commutative Ring

Abstract EN

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors.

Let (Γ(R))c denote the complement of the zero-divisor graph Γ(R) of R.

It is shown that if (Γ(R))c is connected, then its radius is equal to 2 and we also determine the center of (Γ(R))c.

It is proved that if (Γ(R))c is connected, then its girth is equal to 3, and we also discuss about its girth in the case when (Γ(R))c is not connected.

We discuss about the cliques in (Γ(R))c.