Characterization of multiplicative Lie triple derivations on rings
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.
scientific article; zbMATH DE number 7022987
scientific article