Ad-nilpotent Elements of Semiprime Rings with Involution

Abstract

AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.