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

Abstract

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 ${\rho }_1={\rho }_2$.

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