AIMS Mathematics, 2020, 5(4): 3472-3479. doi: 10.3934/math.2020225

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On an identity involving generalized derivations and Lie ideals of prime rings

Department of Mathematics, Patel Memorial National College, Rajpura-140401, Punjab, India

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 $F=0.$
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