## Characterization of multiplicative Lie triple derivations on rings.(English)Zbl 1474.16113

Summary: Let $$\mathcal{R}$$ be a ring having unit 1. Denote by $$\mathcal{Z}$$$$\left(\mathcal{R}\right)$$ the center of $$\mathcal{R}$$. Assume that the characteristic of $$\mathcal{R}$$ is not 2 and there is an idempotent element $$e \in$$$$\mathcal{R}$$ such that $$a \mathcal{R} e$$$$= \left\{0\right\} \Rightarrow a = 0 \text{ and } a \mathcal{R} \left(1 - e\right) = \left\{0\right\} \Rightarrow a = 0$$. It is shown that, under some mild conditions, a map $$L : \mathcal{R} \rightarrow \mathcal{R}$$ is a multiplicative Lie triple derivation if and only if $$L \left(x\right) = \delta \left(x\right) + h \left(x\right)$$ for all $$x \in \mathcal{R}$$, where $$\delta : \mathcal{R} \rightarrow \mathcal{R}$$ is an additive derivation and $$h : \mathcal{R} \rightarrow \mathcal{Z} \left(\mathcal{R}\right)$$ is a map satisfying $$h \left(\left[\left[a, b\right], c\right]\right) = 0$$ for all $$a, b, c \in \mathcal{R}$$. As applications, all Lie (triple) derivations on prime rings and von Neumann algebras are characterized, which generalize some known results.

### MSC:

 16W25 Derivations, actions of Lie algebras
Full Text:

### References:

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