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Are you also looking for holomorphic manifolds with $\dim \mathcal O=\infty$?

In that case, in the paper by Sasane On the Krull Dimension of Rings of Transfer Functions [Acta Applicandae Mathematicae Volume 103, Number 2 , (2008), 161-168] it is shown that the Krull dimension of $\mathcal{O}(\Omega)$ is infinite for any nonempty open subset $\Omega$ of $\mathbb{C}$ (see Corollary 2.3).

In particular the ring of entire functions $\mathcal{O}(\mathbb{C})$ has infinite Krull dimension.

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Are you also look looking for holomorphic manifolds with $\dim \mathcal O=\infty$?

In that case, in the paper by Sasane On the Krull Dimension of Rings of Transfer Functions [Acta Applicandae Mathematicae Volume 103, Number 2, 161-168] it is shown that the Krull dimension of $\mathcal{O}(\Omega)$ is infinite for any nonempty open subset $\Omega$ of $\mathbb{C}$ (see Corollary 2.3).

In particular the ring of entire functions $\mathcal{O}(\mathbb{C})$ has infinite Krull dimension.

1

Do you also look for holomorphic manifolds with $\dim \mathcal O=\infty$?

In that case, in the paper by Sasane On the Krull Dimension of Rings of Transfer Functions [Acta Applicandae Mathematicae Volume 103, Number 2, 161-168] it is shown that the Krull dimension of $\mathcal{O}(\Omega)$ is infinite for any nonempty open subset $\Omega$ of $\mathbb{C}$ (see Corollary 2.3).

In particular the ring of entire functions $\mathcal{O}(\mathbb{C})$ has infinite Krull dimension.