Left ideals of polynomial rings in several indeterminates

Abstract

We consider maximal left ideals L of the polynomial ring R[λ1, …, λn], for R noncommutative. In §1 we reprove and generalize Resco's result that any maximal left ideal L is generated by ≤ n elements whenever R is simple Artinian, and obtain more precise information about the generators when R satisfies a PI. In many instances, fewer than n generators suffice; this is considered in §3, by means of various examples. In §2 we see by a straightforward argument that L has bounded height as a prime left ideal whenever R is a simple Pl-ring, but this does not- hold in general for R simple Artinian.