On the complement of the total zero-divisor graph of a commutative ring

Abstract

The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. Let Z(R) denote the set of all zero-divisors of R and let us denote $Z(R)\backslash \{0\}$ by $Z(R)^{*}$. Let R be a ring such that $Z(R)^{*}\ne \emptyset$. With R, Alen Duric et al. (J Algebra Appl, 2019. https://doi.org/10.1142/S0219498819501901) introduced and investigated an undirected graph denoted by ZT(R) whose vertex set is $Z(R)^{*}$ and distinct vertices x and y are adjacent in ZT(R) if and only if $xy = 0$ and $x + y\in Z(R)$. Let $(ZT(R))^{c}$ denote the complement of ZT(R). In this article, we determine when $(ZT(R))^{c}$ is connected and in the case $(ZT(R))^{c}$ is connected, we determine the diameter and radius of $(ZT(R))^{c}$.