D. Feldman

Non-isomorphism of rings of integer-coefficient holomorphic functions on disks of varying radius

For $\rho \in (0,1]$, let $R(\rho) = \mathbb{Z}[[z]] \cap O(B(0,\rho))$ denote the ring of power series with integer Taylor coefficients converging on the open disk $B(0,\rho)$. 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 $\mathbb{Z}$, 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 $|w| = \rho_1$ forces the image $g(B(0,\rho_2))$ into $B(0,\rho_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $\rho_1 = \rho_2$.