Idempotent values of commutators involving generalized derivations

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.