Abstract:
In the present article, we characterize generalized derivations and left multipliers of prime rings involving commutators with idempotent values. Precisely, we prove that if a prime ring of characteristic different from $2$ admits a generalized derivation $G$ with an associative nonzero derivation $g$ of $R$ such that $[G(u),u]^{n}=[G(u),u]$ for all $u\in\{[x,y]:x,y\in L\},$ where $L$ a noncentral Lie ideal of $R$ and $n>1$ is a fixed integer, then one of the following holds:
$R$ satisfies $s_{4}$ and there exists $\lambda\in C,$ the extended centroid of $R$ such that $G(x)=ax+xa+\lambda x$ for all $x\in R,$ where $a\in U,$ the Utumi quotient ring of $R,$
there exists $\gamma\in C$ such that $G(x)=\gamma x$ for all $x\in R.$
As an application, we describe the structure of left multipliers of prime rings satisfying the condition $([T^m (u),u] )^{n}=[T^m (u),u]$ for all $u\in \{[x,y]: x,y\in L\},$ where $m,n>1$ are fixed integers. In the end, we give an example showing that the hypothesis of our main theorem is not redundant.
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