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 ΓE(R) and the associate class graph ΓA(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), ΓA(R) and ΓE(R). We prove that the count for ΓA(R) is bounded below by the count for ΓE(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 ΓA(R)≅ΓE(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)≅ΓE(R) if and only if R≅ ℤ2 n for some n≥2, or R≅ℤ4, or R≅ℤ2[x]/(x2).
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BibTeX
@misc{kadu2026zero,
title = {{Zero divisors of Gorenstein Rings}},
author = {Ganesh S. Kadu and Vishnu Tanpure},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36021},
eprint = {2609.36021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36021},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zero divisors of Gorenstein Rings
AU - Ganesh S. Kadu
AU - Vishnu Tanpure
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36021
UR - https://arxiv.org/abs/2609.36021
AB - 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)≅Γ_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)≅Γ_E(R) if and only if R≅ ℤ_2^ n for some n≥2, or R≅ℤ_4, or R≅ℤ_2[x]/(x^2).
ER -