Differential commutator identities

AbstractI.N. Herstein proved that if R is a prime ring satisfying a differential identity [x1d,x2d]=E2(x1d,x2d), with d a nonzero derivation of R, then R embeds isomorphically in M2(F) for F a field. We consider a natural generalization of this result for the class of polynomials En(X)=[En-1(x1,…,xn-1),xn]. Using matrix computations, we prove that if R satisfies a differential identity En(x1d1,…,xndn), or [En(…,xidi,…),Em(…,yjhj,…)] with some restrictions, then R must embed in M2(F), but that differential identities using [[En,Em],Es] with m,n,s>1 need not force R to embed in M2(F). These results hold if the expressions are identities for a noncommutative Lie ideal of R, rather than for R itself