Commutative Boolean monoids, reduced rings, and 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 whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=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) is the (undirected) graph whose vertices are the equivalence classes induced by ∼ other than [0] and [1], such that distinct vertices [r] and [s] are adjacent in ΓE(R) if and only if rs=0. We investigate ΓE(R) when R is reduced and are interested in when ΓE(R)≅Γ(S) for a reduced ring S. Among other results, it is shown that ΓE(R)≅Γ(B) for some Boolean ring B if and only if Γ(R) (and hence ΓE(R)) is a complemented graph, and this is equivalent to the total quotient ring of R being a von Neumann regular ring.