Simple algebras of Weyl type

Over a field F of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector space A[D] = A circle times F[D] from any pair of a commutative associative algebra A with an identity element and the polynomial algebra F[D] of a commutative derivation subalgebra D of F. We prove that A[D], as a Lie algebra (module its center) or as an associative algebra, is simple if and only if A is D-simple and, A[D] acts faithfully on A. Thus we obtain a lot of simple algebras.