Structure of Weyl type Lie algebras

Let A be a unital commutative associative algebra over a field F of characteristic zero, D a commutative subalgebra of DerF(A) (all derivations of the associative algebra A). We assume that A is D-simple and denote the center of the Weyl type algebra A[D] by F1 which is an extension field of F when A[D] is simple. In this paper, it is proved that the simple associative algebras A[D] are noncommutative domains, and then the derivations of the simple associative algebras A[D] and of the associated Lie algebras A[D]L are completely determined when dimF1F1D<∞.