Characterization of Generating Ideals in Some Rings of Entire Functions

Let $E$ be a ring of entire functions on $\mathbb C^N$ with the operation of pointwise multiplication, and let $f_1,\dots,f_m$ be a set of nonzero elements in $E$. The ideal $E(f_1,\dots,f_m)$ in $E$ with generators $f_1,\dots,f_m$ is said to be generating if $E(f_1,\dots,f_m) = E$. The generating ideals in rings of entire functions on $\mathbb C^N$ determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of $d(z):=\max_{1\le j\le m}d_j(z)$, where $d_j(z)$ is the distance from a point $z\in\mathbb C^N$ to the zero set of $f_j$, $1\le j\le m$. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of $d(z)$) cannot be given.