The weakly zero-divisor graph of a commutative ring

Mohammad Javad Nikmehr,Abdolreza Azadi,Reza Nikandish

doi:10.33044/revuma.1677

Abstract

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph WΓ(R) with vertex set Z(R) ∗, and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0. It follows that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of WΓ(R) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of WΓ(R) is studied.

Highlights

The theory of graphs associated with rings was started by Beck [9] in 1981 and has grown a lot since
Anderson and Livingston [2] modified Beck’s definition and introduced the notion of zero-divisor graph. This is the most important graph associated with a ring, and zero-divisor graphs and various generalizations of it have attracted many researchers; see for instance [1, 7, 13, 8, 5, 4, 10, 16, 17]
The ring R is said to be reduced if it has no non-zero nilpotent element

Summary

Introduction

The theory of graphs associated with rings was started by Beck [9] in 1981 and has grown a lot since . The zero-divisor graph of a ring R, denoted by Γ(R), is a graph with the vertex set Z(R)∗, and two distinct vertices x and y are adjacent if and only if xy = 0. The weakly zero-divisor graph of R is defined as the graph W Γ(R) with the vertex set Z(R)∗ = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0.

Results

Conclusion