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

Abstract

For $ρ\in (0,1]$, let $R(ρ) = \mathbb{Z}[[z]] \cap O(B(0,ρ))$ 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 $\mathbb{Z}$, 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 $|w| = ρ_1$ forces the image $g(B(0,ρ_2))$ into $B(0,ρ_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $ρ_1 = ρ_2$.

Disclosure

“ρ elementarily equivalent? (ii) Do there exist ring homomorphisms R(ρ1 ) → R(ρ2 ) for ρ1 ̸= ρ2 ? (iii) Is R(ρ) a Bézout domain? Declaration on the Use of Artificial Intelligence In preparing this paper, the authors made use of AI language model as- sistance (Claude) as a tool in the writing process—including LATEX prepa- ration and editing—and in exploratory work, such as proof-checking and verifying arguments. All mathematical content and results have been independently reviewed”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:05:31.283880Z

Structural counts