Answer to Question #23567 in Abstract Algebra for jeremy

Question

For a finite-dimensional k-algebra R, let T(R) = rad R + [R,R], where [R,R] denotes the subgroup of R generated by ab − ba for all a, b ∈ R. Assume that k has characteristic p > 0. Show that T(R) ⊆ {a ∈ R : a^p^m ∈ [R,R] for some m ≥ 1}, with equality if k is a splitting field for R.