I want to prove that the set of entire functions are not a noetherian ring.
My attempt:
A noetherian ring is a ring, for which there is an ascending sequence of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq ... $ such that there exists an $n\in \mathbb{N}: I_n = I_{n+1} = ...$
We can create a complex sequence $(z_n)_{n\in \mathbb{N}}$, $z_0\neq 0$, $\lim_{n\rightarrow \infty}| z_n| = \infty$ which has no limit point in $\mathbb{C}$. Furthermore, we restrict recurring values, meaning $z_k \neq z_l$ for $k\neq l$.
Then we can apply Weierstrass factorization theorem.
Therefore there exists an entire function
$$f(z) = z^m \prod^{\infty}_{n=1} E_{r_n}(\frac{z}{z_n})$$
that has its roots exactly at $(z_n)_{n\in \mathbb{N}}$.
Define: $I_n = \{f \in H(\mathbb{C}) \mid f(z_k) = 0 \text{ for } k\geq n\}$. This set is clearly a subgroup of the set of entire functions (which are obviously a ring). Also, $\forall z\in \mathbb{C} \, \forall f\in I_n: z\cdot f \in I_n$, making it an ideal.
\begin{align*}&\Rightarrow I_1\subsetneq I_2 \subsetneq I_3 \subsetneq \\ &\Rightarrow \nexists n\in \mathbb{N}: I_n=I_{n+1}=... \end{align*}
which proves the set of entire functions is not a noetherian ring.
Is my proof correct? Is there anything to add?