Simple Lie algebras of special type

Let K be a field, let A be an associative, commutative K-algebra, and let D elta be a nonzero K vector space of commuting X-derivations of A. Then, wit h a rather natural definition, W(A, Delta) = A x(K) Delta = A Delta becomes a Lie algebra, a Witt type algebra. In addition, there is a map div: W(A, Delta) --> A called the divergence and its kernel S = S(A, Delta) is a Lie subalgebra, a special type algebra, in this paper, we study S from a ring t heoretic point of view, obtaining sufficient conditions for the Lie simplic ity of [S, S]. While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermo re, some of the preliminary lemmas are of interest in their own right and m ay, in time, lead to a more satisfactory answer. (C) 2000 Academic Press.