Multiplicative Lie derivation of triangular 3-matrix rings

Abstract: A map $\phi$ on an associative ring is called a multiplicative Lie derivation if $\phi([x,y])=[\phi(x),y]+[x,\phi(y)]$ 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 $\varphi:\mathcal T\to\mathcal T$ has the standard form $\varphi=\delta+\gamma$ with $\delta $ a derivation and $\gamma$ a center valued map vanishing each commutator.