Derivations satisfying certain algebraic identities on Lie ideals

Gurninder S. Sandhu, Deepak Kumar

Let d be a derivation of a semiprime ring R and L a nonzero Lie ideal of R. In this note, it is proved that every noncentral square-closed Lie ideal of R contains a nonzero ideal of R. Further, we use this result to characterize the conditions: $d(xy) = d(x)d(y), d(xy) = d(y)d(x)$ on L. With this, a theorem of Ali et al. [14] can be deduced.