Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceAD] =A?FD] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraFD] of a commutative derivation subalgebraD ofA We prove thatAD], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andAD] acts faithfully onA. Thus we obtain a lot of simple algebras.