Isomorphisms and automorphisms of minimal comaximal hypergraphs of commutative rings

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.