Isomorphisms and automorphisms of minimal comaximal hypergraphs of commutative rings

Abstract

Let $R$ be a commutative ring with exactly $n\ge 2$ maximal ideals and let $k\ge 2$. We study the minimal $k$-uniform comaximal hypergraph $\mathcal{H}_k(R)$, whose vertices are the nonunits outside the Jacobson radical and whose edges are the irredundant $k$-element comaximal families. For $2\le k\le n$ we show that $\mathcal{H}_k(R)$ is a blow-up of the hypergraph of irredundant $k$-coverings of $\{1,\dots,n\}$, 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 $\mathrm{Aut}(\mathcal{H}_k(R))$ and characterize isomorphisms of such hypergraphs. For $3\le k\le n$ its automorphism group properly contains that of the comaximal graph, and for finite rings with $n\ge 3$ and every $2\le k\le n$, $\mathcal{H}_k(R)$ 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.