D-SIMPLE RINGS AND PRINCIPAL MAXIMAL IDEALS OF THE WEYL ALGEBRA

We prove that if the order-one differential operator S=(partial)1 + sumi=2^n (beta)i(partial)i + (gamma), with (beta)i,(gamma) in K[x1,xn], generates a maximal left ideal of the Weyl algebra An(K), then S does not admit any Darboux differential operator in K[x1,xn]langle partial_2,ldots,partial_nrangle ; hence in particular, the derivation )partial)1 + sumi=2^ n beta_ipartial_i does not admit any Darboux polynomial in K[x1,xn]. We show that the converse is true when (beta)i in K[x1,xi], for every i=2,ldots,n. Then, we generalize to K [x1,xn] the classical result of Shamsuddin that characterizes the simple linear derivations of K[x1,x2]. Finally, we establish a criterion for the left ideal generated by S in An(K) to be maximal in terms of the existence of polynomial solutions of a finite system of differential polynomial equations.