Abstract
Let \({\mathcal{R}}\) be a 2-torsion free commutative ring with identity and \({{\rm M}_n(\mathcal{R}) (n\geq 2)}\) be the full matrix algebra over \({\mathcal{R}}\). In this note, we prove that every nonlinear Lie triple derivation on \({{\rm M}_n(\mathcal{R})}\) is of the standard form, i.e. it can be expressed as a sum of an inner derivation, an additive induced derivation and a functional annihilating all second commutators of \({{\rm M}_n(\mathcal{R})}\). A open conjecture about Lie n-derivations is posed at the end of this note.
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Communicated by J. S. Wilson.
The work of the first author is supported by a research foundation of Huaqiao University (Grant No. 10BS323). The work of the second author is partially supported by the National Natural Science Foundation of China (Grant No. 10871023).
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Xiao, Z., Wei, F. Nonlinear lie-type derivations on full matrix algebras. Monatsh Math 170, 77–88 (2013). https://doi.org/10.1007/s00605-012-0388-7
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DOI: https://doi.org/10.1007/s00605-012-0388-7