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Multiplicative Lie derivations on triangular \(n\)-matrix rings. (English) Zbl 1497.16043

Summary: Let \(\mathcal{T}\) be a triangular \(n\)-matrix ring \((n \geq 2)\). It is shown that, under some mild assumptions, a map \(\delta : \mathcal{T} \to \mathcal{T}\) is a multiplicative Lie derivation if and only if \(\delta(X) = d(X) + \gamma(X)\) holds for all \(X \in \mathcal{T}\), where \(d : \mathcal{T} \to \mathcal{T}\) is an additive derivation and \(\gamma : \mathcal{T} \to \mathcal{Z}(\mathcal{T})\) is a central-valued map vanishing on all commutators.

MSC:

16W25 Derivations, actions of Lie algebras
16S50 Endomorphism rings; matrix rings
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