Let $D$ be a polydisc in $\mathbb{C}^n$ and $\mathcal{O}_D$ the sheaf of holomorphic functions on $D$. Further let $J\subset \mathcal{O}_D$ be a finitely generated ideal, since $D$ is Stein it follows that $J\left(D\right)$ is also finitely generated. Suppose that $\mathcal{O}_D$ is equipped with the topology of uniform convergence on compact sets. I am wondering if $J(D)$ is then closed in $\mathcal{O}_D(D)$.
When $J$ is the ideal associated to a subvariety $A$, i.e. $J=\left\{f\in\mathcal{\mathcal{O}_D}\mid f\vert_A=0\right\}$, then I can show that it has to be closed.
More concretely the question is essentially: Suppose $\left\{a_i\right\}_{i=1}^p$ are holomorphic functions on $D$ and the sequence $a_1\cdot f^j_1 + \dots + a_p \cdot f^j_p$ converges uniformly to $g$, then there exist $g_i$ holomorphic such that
$g=a_1 \cdot g_1 + \dots + a_p\cdot g_p.$
I want to prove this to obtain a Frechet topology on the structure sheaf of (non-reduced) complex analytic spaces, which is compatible with the quotient topology $\mathcal{O}_D/ J$.
Yes, this is true. A proof can be found in "Analytic Functions of Several Complex Variables" by R. Gunning and H. Rossi on pages 82 - 85, in particular Theorem 3 in section D of chapter 2.