Prime Rings with Generalized Derivations
Received:May 12, 2006  Revised:October 12, 2006
Key Words: prime ring   Lie ideal   generalized derivation.
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 Author Name Affiliation HUANG Shu-liang Department of Mathematics, Chuzhou University, Ahnui 239012, China School of Mathematics and Computer Science, Nanjing Normal University, Jiangsu 210097, China FU Shi-tai School of Mathematics and Computer Science, Nanjing Normal University, Jiangsu 210097, China
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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 $u,v\in U$.