| Popis: |
Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R,$ $C$ the extended centroid of $R$ and $L$ a noncentral Lie ideal of $R.$ If $R$admits a generalized derivation $F$ associated with a derivation $\delta$ of $R$ such that for some fixed integers $m,n\geq 1,$ $F([u,v])^{m}=[u,v]_{n}$ for all $u,v\in L,$ then one of the following holds true: (i) $R$ satisfies $s_{4},$ the standard identity in four variables. (ii) there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R.$ Moreover, if $n=1,$ then $\lambda^{m}=1$ and if $n>1,$ then $F=0.$ |
