Multiplicative Lie derivation of triangular 3-matrix rings

Abstract

A map $\varphi$ on an associative ring is called a multiplicative Lie derivation if $\varphi ([x,y])=[\varphi (x),y]+[x,\varphi (y)]$ holds for any elements $x,y$, where $[x,y]=xy-yx$ is the Lie product. In the paper, we discuss the multiplicative Lie derivations on the triangular 3-matrix rings $\mathcal{T}={\mathcal{T}}_3({\mathcal{R}}_i;{\mathcal{M}}_{ij})$. Under the standard assumption $Q_i\mathcal{Z}(\mathcal{T})Q_i=\mathcal{Z}(Q_i\mathcal{T}{Q}_i)$, $i=1,2,3$, we show that every multiplicative Lie derivation $\phi :\mathcal{T}\to \mathcal{T}$ has the standard form $\phi =\delta +\gamma$ with $\delta$ a derivation and $\gamma$ a center valued map vanishing each commutator.

23pages

References in corpus (1)

• Characterization of Lie Derivations on von Neumann Algebras