Isomorphisms and automorphisms of minimal comaximal hypergraphs of commutative rings
Abstract
Let R be a commutative ring with exactly n≥ 2 maximal ideals and let k≥ 2. We study the minimal k-uniform comaximal hypergraph ℋk(R), whose vertices are the nonunits outside the Jacobson radical and whose edges are the irredundant k-element comaximal families. For 2≤ k≤ n we show that ℋk(R) is a blow-up of the hypergraph of irredundant k-coverings of {1,…,n}, and that its reduced form reconstructs the maximal ideals up to permutation; for k>n there are no edges. We obtain a general description of Aut(ℋk(R)) and characterize isomorphisms of such hypergraphs. For 3≤ k≤ n its automorphism group properly contains that of the comaximal graph, and for finite rings with n≥ 3 and every 2≤ k≤ n, ℋk(R) has the same isomorphism invariants as the comaximal graph, namely |J(R)| and R/J(R). The case k=2 recovers and extends the automorphism results for comaximal graphs obtained previously. We also compare the minimal and non-minimal comaximal hypergraphs and identify a condition under which the latter has the same automorphism group and isomorphism invariants as the comaximal graph. The same method determines the vertex set and the automorphism group of the k-zero-divisor hypergraph of a product of integral domains, and shows that for finite products of finite fields this hypergraph determines the ring.
Related
- Domination and maximal intersecting families in comaximal graphs of commutative rings
- Automorphisms of Twisted Chevalley Groups over Commutative Rings
- Finite Non-Commutative Centrally Essential Rings of Minimal Order
- Regularity of profinite isomorphisms of hyperbolic 3-manifolds
- Identities and isomorphisms of finite graded matrix algebras
BibTeX
@misc{mehdinezhad2026isomorphisms,
title = {{Isomorphisms and automorphisms of minimal comaximal hypergraphs of commutative rings}},
author = {Elham Mehdi-Nezhad},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07992},
eprint = {2611.07992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07992},
}
RIS
TY - UNPB
TI - Isomorphisms and automorphisms of minimal comaximal hypergraphs of commutative rings
AU - Elham Mehdi-Nezhad
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07992
UR - https://arxiv.org/abs/2611.07992
AB - Let R be a commutative ring with exactly n≥ 2 maximal ideals and let k≥ 2. We study the minimal k-uniform comaximal hypergraph ℋ_k(R), whose vertices are the nonunits outside the Jacobson radical and whose edges are the irredundant k-element comaximal families. For 2≤ k≤ n we show that ℋ_k(R) is a blow-up of the hypergraph of irredundant k-coverings of {1,…,n}, and that its reduced form reconstructs the maximal ideals up to permutation; for k>n there are no edges. We obtain a general description of Aut(ℋ_k(R)) and characterize isomorphisms of such hypergraphs. For 3≤ k≤ n its automorphism group properly contains that of the comaximal graph, and for finite rings with n≥ 3 and every 2≤ k≤ n, ℋ_k(R) has the same isomorphism invariants as the comaximal graph, namely |J(R)| and R/J(R). The case k=2 recovers and extends the automorphism results for comaximal graphs obtained previously. We also compare the minimal and non-minimal comaximal hypergraphs and identify a condition under which the latter has the same automorphism group and isomorphism invariants as the comaximal graph. The same method determines the vertex set and the automorphism group of the k-zero-divisor hypergraph of a product of integral domains, and shows that for finite products of finite fields this hypergraph determines the ring.
ER -