I saw this statement mentioned the other day, and I have some questions about what we are actually saying. What do we mean by a "locally holomorphic function around a fixed point in $\mathbb{C}$?" My interpretation of it would be to fix some $z \in \mathbb{C}$, and say that we are looking at functions $f$ such that $f$ is holomorphic on some open set $\Omega$, where $z \in \Omega$. But we could also mean it where, instead of taking open sets, we could take the more restrictive $\epsilon$-neighborhoods around $z$. I've heard "locally" being used in both these contexts before.
There's also some ambiguity around the phrase "fixed point." I think it means what I say above, but it could also mean we are looking at functions $f$ that fix some $z_0 \in \mathbb{C}$, $f(z_0) = z_0$.
And after this is clarified, how would we begin to go about proving that this ring is a PID? From similar questions asked on this site, it seems that where the function is $0$ plays an important role in determining ideals, but I'm not exactly sure of the specifics involved.
Any help would be greatly appreciated!