2026/07/24 by Jon Bannon, David Feldman
#math.RA #math.CV #math.NT
For ρ∈ (0,1], let R(ρ) = ℤ[[z]] ∩ O(B(0,ρ)) denote the ring of power series with integer Taylor coefficients converging on the open disk B(0,ρ). We prove that these rings are pairwise non-isomorphic as abstract rings. Three ingredients drive the proof: the ideal (z) is the unique principal ideal with quotient ℤ, so any isomorphism sends z to a generator g of (z); every isomorphism is substitution by g, because it respects the (z)-adic filtration; and a Hadamard gap series with a natural boundary at |w| = ρ1 forces the image g(B(0,ρ2)) into B(0,ρ1), after which the Schwarz lemma and integrality of coefficients force g = ± z and ρ1 = ρ2.