Ideal of Ring of holomorphic functions

Question

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.

Answer 1

For example the set of all holomorphic functions that vanish at $0$ is a nontrivial ideal. More generally any prescribed discrete zero set that is possible for a holomorphic function will work similarly.

Answer 2

Choose a subset $S\subset\Bbb C$. Then you can define $I$ to be the set of holomorphic functions $f$ such that $f(S)=\{0\}$.

Warning: don't choose $S$ too big. If you do, the identity theroem will reduce $I$ to a single function!

Answer 3

Every finitely generated (proper) ideal $I$ of $\mathcal{O}(\mathbb{C})$ has the property that $$ \bigcap_{f\in I} f^{-1}(0) \neq \emptyset. $$ In particular, every finitely generated maximal ideal is of the form $M_p = \{ f : f(p) = 0 \}$.

But there are other ideals. Take for example any discrete sequence $z_j \to \infty$, and let $F_n$ be a function whose zero set is exactly $\{ z_n, z_{n+1}, \ldots \}$. Let $I$ be the ideal generated by $\{ F_n \}_{n=1}^\infty$. Then it is not hard to see that there is no set $Z$ on which all elements of $I$ vanish. In particular, $I$ is contained in some maximal ideal that does not consist of all functions vanishing at a particular point.

See for example On the ideal structure of the ring of entire functions (Henriksen, Pacific J. Math. 2, (1952)) for a lot of information about the ideals in $\mathcal{O}(\mathbb{C})$.