带有广义导子的素环

The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R→ R satisfying δ(xy) = δ(x)y xd(y) for all x,y∈R,where d is a derivation on R.Such a function δis called a generalized derivation.Suppose that U is a Lie ideal of R such that u2 ∈ U for all u ∈U.In this paper,we prove that U(C)Z(R) when one of the following holds:(1)δ(u,v]) = uov (2)δ(u,v]) uov=O(3)δ(uov) =u,v](4)δ(uov) u,v]= O for all u,v ∈U.

Prime Rings with Generalized Derivations

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 $u,v\in U$.