There are two results in Complex Analysis that have a counterpart in Algebra:
-If we consider the ring of holomorphic functions in an open set $\mathcal H(U)$ with the usual sum and product, every finitely generated ideal is principal. In fact it is generated by any holomorphic function that vanishes exactly where the ideal $I$ and with the same multiplicitiy.
This is the same as in $\mathbb C[X]$ (although here all the ideals are finitely generated), where every ideal is characterized by the zeroes and the multiplicities.
-In several complex variables, a function which is holomorphic in $U\setminus\{p\}$ is holomorphic in $U$.
If we restricted to rational functions $\displaystyle \frac{p(z)}{q(z)}$, this would be a corrolary from the fact that $q(z)=0$ has codimension $1$. Hence it can't be a point.
My question is if it is a coincidence that in these two cases, holomorphic functions act somewhat similar to polynomials, or if it is an instance of a more general phenomenon?