Preprint A236/2003
d-simple rings and principal maximal ideals of the Weyl algebra

Jose Carlos SOUZA Jr. | LEQUAIN, Yves | LEVCOVITZ, Daniel

**Keywords: **
simple derivation | maximal left ideal of the Weyl algebra |

Let K be a field of characteristic zero and A_n the nth-Weyl algebra over K. Let d be a derivation of K[X_1,...,X_n]. First, we show that if there exists an element h of
K[X_1,...,X_n] such that d+h generates a maximal left ideal of A_n, then d+h does not admit any Darboux operator in
K[X-1,...,X_n,d/dX_2,...,d/dX_n], hence in particular, d does not admit any Darboux polynomial. Next, we show that if furthermore d is a Shamsuddin derivation, then d is necessarily a simple derivation of K[X_1,...,X_n]. Finally, we use our results to construct new simple derivation of K[X_1,...,X_n] and to recover many results on maximal ideals of the Weyl algebra A_n.