Some remarks on the compressed zero-divisor graph

Abstract

Let R be a commutative ring with 1≠0. The zero-divisor graph Γ(R) of R is the (undirected) graph with vertices the nonzero zero-divisors of R, and distinct vertices r and s are adjacent if and only if rs=0. The relation on R given by r∼s if and only if annR(r)=annR(s) is an equivalence relation. The compressed zero-divisor graph ΓE(R) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [r] and [s] are adjacent if and only if rs=0. Let RE be the set of equivalence classes for ∼ on R. Then RE is a commutative monoid with multiplication [r][s]=[rs]. In this paper, we continue our study of the monoid RE and the compressed zero-divisor graph ΓE(R). We consider several equivalence relations on R and their corresponding graph-theoretic translations to Γ(R). We also show that the girth of ΓE(R) is three if it contains a cycle and determine the structure of ΓE(R) when it is acyclic and the monoids RE when ΓE(R) is a star graph.