Prime Rings with Generalized Derivations

Abstract:

      The concept of derivations and generalized inner derivations has been generalized as an additive function $\delta:R \longrightarrow R$ satisfying $\delta(xy)=\delta(x)y+xd(y)$ for all $x,y\in R$, where $d$ is a derivation on $R$. Such a function $\delta $ is called a generalized derivation. Suppose that $U$ is a Lie ideal of $R$ such that $u^{2}\in U$ for all $u\in U$. In this paper, we prove that $U\subseteq Z(R)$ when one of the following holds: (1) $ \delta([u,v])=u\circ v $ (2) $ \delta([u,v])+u\circ v=0 $ (3) $ \delta(u\circ v)=[u,v] $ (4) $ \delta(u\circ v)+[u,v]=0 $ for all $u,v\in U$.