带有广义导子的素环

Prime Rings with Generalized Derivations

中文关键词:  素环  李理想  广义导子.

英文关键词:prime ring  Lie ideal  generalized derivation.

中文摘要:

      导子和广义导子的概念已经被推广为 $R$上的一个满足$\delta(xy)=\delta(x)y+xd(y)$的函数$\delta$，其中 $d$为$R$的一个导子，这样的函数称为广义导子.假设$U$ 是$R$的一个平方封闭的李理想.本文证明当下列四个条件之一成立时 $U$为中心李理想：(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 $ 对所有的 $u,v\in U$.

英文摘要:

      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 $u,v\in U$.