Consider $X$ a complex manifold. Denote $x\in X$ a point and $O_x$ as the local holomorphic function ring at $x$.
Assume $X$ is not algebraic.
$\textbf{Q1:}$ Is $O_x$ Noetherian? If it is Noetherian, then my guess is that it can be locally defined by equations and it follows from Chow's theorem that $X$ is algebraic as well. Is there something wrong here?
$\textbf{Q2:}$ Is $O$ ring of holomorphic functions over non-compact $X$ Noetherian?
For arbitrary complex manifold, locally, $\mathcal O_{X,x}=\mathcal O_{\mathbb C^n,0}$ is noetherian.