The total zero-divisor graph of commutative rings

Abstract

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of 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 graphs of the ring ${\mathbb{Z}}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb{Z}}_m$ and ${\mathbb{Z}}_n$ are isomorphic if and only if $m=n$.

Cited by in corpus (2)

• Compressed zero-divisor graphs of noncommutative rings
• Categorial properties of compressed zero-divisor graphs of finite commutative rings