AI 中文总结
研究不同半径圆盘上整数系数全纯函数环的非同构性，通过分析理想\((z)\)、\((z)\)进滤层以及哈达玛间隙级数等，证明这些环两两抽象非同构。
AI 中文摘要
对于\(\rho\in(0,1]\)，令\(R(\rho)=\mathbb{Z}[[z]]\cap O(B(0,\rho))\)表示在开圆盘\(B(0,\rho)\)上收敛且泰勒系数为整数的幂级数环。我们证明了这些环作为抽象环是两两非同构的。证明基于三点：理想\((z)\)是唯一商为\(\mathbb{Z}\)的主理想，同构将\(z\)映射到\((z)\)的生成元\(g\)；同构是由\(g\)进行的代换，因为它尊重\((z)\)进滤层；具有在\(|w|=\rho_1\)处自然边界的哈达玛间隙级数迫使像\(g(B(0,\rho_2))\)包含于\(B(0,\rho_1)\)，进而由施瓦茨引理和系数的整数性迫使\(g = \pm z\)且\(\rho_1 = \rho_2\)。
英文摘要
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 $ρ_1 = ρ_2$.
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