Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then $K[T_1,\ldots,T_n]/\mathfrak{m}$ is a finite extension of $K$.

I am interested in a noncommutative version of this theorem. To be more precise: Let $K$ be a field and let $\delta_1,\ldots,\delta_n$ be commuting derivations on $K$. Suppose $\mathfrak{m}$ is a maximal left ideal of $K[\delta_1,\ldots,\delta_n]$. Is $K[\delta_1,\ldots,\delta_n]/\mathfrak{m}$ finite dimensional as a vector space over $K$? If not, is there a simple counterexample?

This seems like a very natural question to ask but I was unable to find anything relevant in the literature. I also tried to lift the proof of the classical theorem to this setting using the technology of associated graded rings/modules but I could not make it work.