Minimal polynomials of algebraic derivations and automorphisms

Let R be a prime ring with extended centroid C. Let a ∈ R be C-algebraic. We compute the minimal polynomial of the inner derivation ad(a) (resp. the inner automorphism Ia for unit a) in terms of the minimal polynomial of a. Chung and Luh proved that the index of nilpotency of a nilpotent derivation is an odd number if char R ≠ 2 and is a 2-power if char R = 2. As an application, we generalize this to arbitrary algebraic derivations and automorphisms.