Zero divisors of Gorenstein Rings

Abstract

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $Γ_E(R)$ and the associate class graph $Γ_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $Γ(R)$, $Γ_A(R)$ and $Γ_E(R)$. We prove that the count for $Γ_A(R)$ is bounded below by the count for $Γ_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $Γ_A(R)\congΓ_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $Γ(R)\congΓ_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.