Minimal polynomials of algebraic derivations and automorphisms

Abstract

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 I a 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.