The total zero-divisor graph of a commutative ring

Abstract

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring [Formula: see text] of integers modulo [Formula: see text] and prove that the total zero-divisor graphs of [Formula: see text] and [Formula: see text] are isomorphic if and only if [Formula: see text].