Certain additive maps on m-power closed Lie ideals

Abstract

Let R be a prime ring with extended centroid C and m a fixed positive integer > 1. A Lie ideal L of R is called m-power closed if \({u^m \in L}\) for all \({u \in L}\) . We prove that if char R = 0 or a prime p > m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dimCRC = 4, m is odd, and \({u^{m-1} \in C}\) for all \({u \in L}\) . Moreover, the additive maps d : L → R satisfying d(um) = mum-1d(u) (resp. d(um) = um-1d(u)) for all \({u \in L}\) are completely characterized if char R = 0 or a prime p > 2(m − 1).