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 R. If D is both a Lie subalgebra and R-submodule of DerkR such that R is D-simple and either char k ≠ 2 or D is not cyclic as an R-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 Δ is a Lie subalgebra of DerkR then we show that R k Δ is simple if and only if R is Δ-simple, the action of R k Δ on R is faithful, and, if char k = 2 and dimk Δ = 1, Δ(R) = R. This generalizes the weaker of two forms of a result by D. S. Passman (1998, J. Algebra34, 682–692), where Δ is abelian. However, a stronger form, in which the action of R k Δ is replaced by that of RΔ k Δ, does not generalize.