Nonlinear generalized Lie n-derivations on triangular algebras

ABSTRACT

Let 𝒯 be a (n−1)-torsion free triangular algebra over a communicative ring ℛ and 𝒵(𝒯) be the center of 𝒯. Let G_L:𝒯→𝒯 be a nonlinear generalized Lie n-derivation with associated Lie n-derivation L. Suppose that $p_n(x_1,x_2,\dots ,x_n)$ is the (n−1)-th commutator defined by n indeterminates $x_1,\dots ,x_n$. In this paper, we prove that under certain assumptions, the nonlinear generalized Lie n-derivation G_L is of the form G_L = δ+τ, where δ:𝒯→𝒯 is an additive generalized derivation on 𝒯 and τ:𝒯→𝒵(𝒯 ) is a mapping vanishing on each (n−1)-th commutator $p_n(x_1,x_2,\dots ,x_n)$. Some relevant applications of this result are presented at the end of this article.

KEYWORDS:

• Nonlinear generalized Lie n-derivation
• triangular algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 15A78

Acknowledgments

The author thanks the referee for her/his careful reading and valuable suggestions to improve the presentation of this paper.