Left ideals of polynomial rings in several indeterminates

We consider maximal left ideals L of the polynomial ring FORMUIA OMITTED 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 PI-ring but this does not hold in general for R simple Artinian.