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.


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
Full Text: DOI


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