Characterizations of Lie-type derivations of triangular algebras with local actions

Abstract

Let $\mathbb {N}$ be the set of nonnegative integers and $\mathfrak {A}$ be a $(n-1)$-torsion free triangular algebra over a commutative ring $\mathcal {R}$. In the present paper, under some mild assumptions on $\mathfrak {A}$, it is prove that if $\delta : \mathfrak { A }\rightarrow \mathfrak { A }$ is an $\mathcal {R}$-linear mapping satisfying $\delta (p_n(X_1, X_2, \cdots , X_n))=\sum \limits _{i=1}^{i=n}p_n(X_1, X_2, \cdots , X_{i-1}, \delta (X_{i}), X_{i+1}, \cdots , X_n)$ for all $X_1, X_2, \cdots , X_n \in \mathfrak {A}$ with $X_1X_2 =0$ (resp. $X_1X_2=P$, where P is a nontrivial idempotent of $\mathfrak {A}$), then $\delta =d+\tau$; where $d:\mathfrak {A}\rightarrow \mathfrak {A}$ is a derivation and $\tau : \mathfrak {A}\rightarrow Z( \mathfrak {A})$ (where $Z( \mathfrak {A})$ is the center of $\mathfrak {A}$) is an $\mathcal {R}$-linear map vanishing at every $(n-1)$-th commutator $p_n(X_1, X_2, \cdots , X_n)$ with $X_1X_2=0$ (resp. $X_1X_2=P$).