## Multiplicative Lie $$n$$-derivations of triangular rings.(English)Zbl 1247.16040

Let $$R$$ be an associative ring and let $$[x,y]=xy-yx$$ denote the Lie product of $$x,y\in R$$. An additive map $$\delta$$ is called a Lie derivation on $$R$$ if it is a derivation with respect to the Lie product. In this paper the concept of Lie derivation is generalized as follows. Let $$p_1(x)=x$$ and, for $$n\geq 2$$, let $$p_n(x_1,\dots,x_n)=[p_{n-1}(x_1,\dots,x_{n-1}),x_n]$$, where $$x,x_1,\dots,x_n\in R$$. A mapping $$\varphi\colon R\to R$$ is a multiplicative Lie $$n$$-derivation if $\varphi(p_n(x_1,\dots,x_n))=\sum_{i=1}^np_n(x_1,\dots,x_{i-1},\varphi(x_i),x_{i+1},\dots,x_n)$ holds for all $$x_1,\dots,x_n\in R$$. For instance, let $$\delta$$ be a Lie derivation on $$R$$ and $$\gamma\colon R\to Z(R)$$ such that $$\gamma(p_n(R,\dots,R))$$ is trivial. Then $$\delta+\gamma$$ is a multiplicative $$n$$-Lie derivation on $$R$$ and multiplicative $$n$$-Lie derivations of this type are said to be of the standard form.
One of the results of the paper gives a necessary and sufficient condition for a multiplicative $$n$$-Lie derivation on an $$(n-1)$$-torsion free triangular ring to be of the standard form. The main theorem gives sufficient conditions on an $$(n-1)$$-torsion free triangular ring for every multiplicative $$n$$-Lie derivation on it being of the standard form.

### MSC:

 16W25 Derivations, actions of Lie algebras 16S50 Endomorphism rings; matrix rings 47B47 Commutators, derivations, elementary operators, etc. 16R60 Functional identities (associative rings and algebras) 15A78 Other algebras built from modules 47L35 Nest algebras, CSL algebras
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