Nonlinear lie-type derivations on full matrix algebras

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.