On the simplicity of Lie algebras of derivations of commutative algebras

Let R be a commutative algebra over a field k. We prove two related results on the simplicity of Lie algebras acting as derivations of A. If D is both a Lie subalgebra and R-submodule of Der, R such that A is D-simple and either char k not equal 2 or D is not cyclic as an A-module or D(R) = R, then we show that D is simple. This extends a previous result from the author (1986, J. London Math. Soc. (2) 33, 33-39) so as to include characteristic 2. If Delta is a Lie subalgebra of Der(k)R then we show that Rx(k)Delta is simple if and only if R is Delta-simple, the action of Rx(k)Delta on R is faithful, and, if char k = 2 and dim(k)Delta = 1, Delta(R) = R. This generalizes the weaker of two forms of a result by D. S. Passman (1998, J. Algebra 34, 682-692), where Delta is abelian. However, a stronger form, in which the action of Rx(k)Delta is replaced by that of R(Delta)x(k)Delta, does not generalize, (C) 2000 Academic Press.