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 $x,yin R$, 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 $uin U$. In this paper, we prove that $Usubseteq Z(R)$ when one of the following holds: (1) $ delta([u,v])=ucirc v $ (2) $ delta([u,v])+ucirc v=0 $ (3) $ delta(ucirc v)=[u,v] $ (4) $ delta(ucirc v)+[u,v]=0 $ for all $u,vin U$.