I´m trying to figure out some algebraic properties of the ring of the entire functions, so I came across with the following problem that I cannot solve.
For any $n\geq 1$, consider the entire function defined as: $$g_{n}(z)=\prod_{k=n}^{\infty}\bigg(1-\frac{z^{2}}{k^{2}}\bigg), z\in \mathbb{C}$$
How can I show that the ideal generated by this functions in the ring of entire functions $\{fg_{n}: f\in H(\mathbb{C}), n\geq 1 \}$ and that moreover this ideal is not principal?
On
In my comment, I showed that $I=\{fg_n | f \in H(\mathbb{C}), n \ge 1\}$ was indeed the required ideal. Let us assume it is finitely generated by some $h_1,...,h_n$.
Note that for all $f \in I$ there exists an integer $N$ such that all integers greater than $N$ are roots of $f$.
Since $h_1,...,h_n$ are in $I$, there must be an integer $N$ such that all integers greater than it are roots of all the $h_i$.
Since the $h_i$ are generators, all integers greater than $N$ are roots of all the functions in $I$. But $N+1$ is not a root of $g_{N+2} \in I$. Contradiction!
This ideal is not even finitely generated. If it were, then it would be generated by $g_1,\ldots,g_N$ for some $N$, But $g_{N+1}$ is not in the ideal generated by $g_1,\ldots,g_N$, since $g_{N+1}(N)\ne0$ but $g_j(N)=0$ for $j\le N$.