Closed ideals with the nulldimensional root set in certain rings of holomorphic functions

Closed ideals $J$ with null-dimensional set of roots are considered in some rings $H$ of functions holomorphic in a pseudoconvex domain $\Omega\subseteq\mathbb C^n$. It is proved (under additional assumptions) that $J$ coincides with the set of functions $f\in H$ with the germs (in all points $z\in\Omega$) are generated over the rings of series convergent at $z$ by the germs of elements of $J$.