Multiplicative Lie-Type Derivations on Rings

Let $\mathcal{R}$ be a ring with the center $\mathcal{Z}(\mathcal{R})$ containing a nontrivial idempotent. Suppose $p_n(X_1,X_2,\cdots, X_n)$ is the polynomial defined by $n$ noncommuting indeterminates $X_1, \cdots, X_n$ and their multiple Lie products. In this article, under a lenient condition on $\mathcal{R}$, it is shown that if a mapping $L : \mathcal{R} \rightarrow \mathcal{R}$ satisfies $L(p_n(A_{1},A_{2},\cdots,A_{n}))= \sum_{k=1}^n p_n(A_1,\cdots, A_{k-1}, L(A_k), A_{k+1},\cdots, A_n)$, for all $A_{1},A_{2},\cdots,A_{n} \in \mathcal{R}$ and $n \geq 2$ be a fixed positive integer, then for all $A,B \in \mathcal{R}$ there exists $Z_{A,B}$ (depending on $A$ and $B$) in $\mathcal{Z}(\mathcal{R})$ such that $L(A+B)=L(A)+L(B)+Z_{A,B}$.