Skew Derivations With Power Central-Values On Commutators

Abstract

In the thesis we study the problem concerning skew derivations with power central-values on commutators in prime rings. Precisely, we prove the following two main results. Main Theorem 1. Let R be a prime ring with charR 6= 2, an automorphism of R, and d a -derivation of R. If [d([x; y]); [x; y]]n = 0 for all x; y 2 R, then either d = 0 or R is a commutative ring. Main Theorem 2. Let R be a prime ring, with an automorphism , and d a nonzero -derivation of R. Suppose that [d(x); x]n 2 Z(R) for all x 2 R. If either charR = 0 or charR > n, then dimC RC 4. As a corollary to Main Theorem 1, we have the following result. Corollary. Let R be a prime ring with charR 6= 2, an automorphism of R, L a noncentral Lie ideal of R, and d a -derivation of R. If [d(x); x]n = 0 for all x 2 L, then either d = 0 or R is a commutative ring. 2000 Mathematics Subject Classi cation. 16R53, 16R60, 16N60. Key words and phrases. Automorphism, prime ring, Martindale quotient ring, skew derivation, M-inner.