[@Wan1]. For example, any exponential function $\lambda(x)$ satisfies the above inequalities. In this case the ring is particularly interesting because it arises naturally from the study of unit root F-crystals from geometry, see Dwork-Sperber [@DS] and Wan [@Wan2] for further discussions.

The first condition, $\lambda(x) + \lambda(y) \leq \lambda(x+y)$, is a natural assumption because it ensures that elements of the form $(1-XY)$ are invertible, a vital condition to this paper. If $\lambda$ does not grow at least as fast as linear, then $(1-XY)^{-1} = 1 + \sum_{i=1}^{\infty} X^kY^k$ is not an element of $R[X; Y, \lambda]$. It is not clear, however, if the second condition, $\lambda(x+y) \leq \lambda(x)\mu(y)$, can be dropped. In fact, we have the following open question from Wan [@Wan1].

Let $R$ be a noetherian ring. Let $\lambda(x)$ be a growth function satisfying $\lambda(x) + \lambda(y) \leq \lambda(x+y)$. Is the intermediate ring $R[X; Y, \lambda]$ always noetherian?

This question is solved affirmatively in this paper if $R$ is a field and there is only one $Y$ variable.

Throughout this paper we assume that $R=F$ is a field, and that $\lambda$ grows at least as fast as linear, i.e. $\lambda(x) + \lambda(y) \leq \lambda(x+y)$ for all $x, y \geq 