Multiplicative Lie-Type Derivations on Rings

Abstract

Let ${\mathcal {R}}$ be a ring with the center ${\mathcal {Z}}({\mathcal {R}})$ containing a nontrivial idempotent. Suppose $p_n(X_1,X_2,\dots , X_n)$ is the polynomial defined by n noncommuting indeterminates $X_1, \dots , X_n$ 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 $L(p_n(A_{1},A_{2},\dots ,A_{n}))= \sum _{k=1}^n p_n(A_1,\dots , A_{k-1}, L(A_k), A_{k+1},\dots , A_n)$, for all $A_{1},A_{2},\dots ,A_{n} \in {\mathcal {R}}$ and $n \ge 2$ 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 $L(A+B)=L(A)+L(B)+Z_{A,B}$.