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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2020; 60(4): 683-710

Published online December 31, 2020

### Characterizations of Lie Triple Higher Derivations of Triangular Algebras by Local Actions

Mohammad Ashraf, Mohd Shuaib Akhtar* and Aisha Jabeen

Department of Mathematics, Aligarh Muslim University, Aligarh, India
e-mail : mashraf80@hotmail.com, mshuaibakhtar@gmail.com and ajabeen329@gmail.com

Received: April 24, 2019; Accepted: February 25, 2020

### Abstract

Let $N$ be the set of nonnegative integers and $A$ be a 2-torsion free triangular algebra over a commutative ring $R$ . In the present paper, under some lenient assumptions on $A$ , it is proved that if $Δ = { δ n } n ∈ ℕ$ is a sequence of $R$ -linear mappings $δ n : A → A$ satisfying $δ n ( [ [ x , y ] , z ] ) = ∑ i + j + k = n [ [ δ i ( x ) , δ j ( y ) ] , δ k ( z ) ]$ for all $x , y , z ∈ A$ with $x y = 0$ (resp. $x y = p$ , where p is a nontrivial idempotent of $A$ ), then for each $n ∈ ℕ$ , $δ n = d n + τ n$ ; where $d n : A → A$ is $R$ -linear mapping satisfying $d n ( x y ) = ∑ i + j = n d i ( x ) d j ( y )$ for all $x , y ∈ A$ , i.e. $D = { d n } n ∈ ℕ$ is a higher derivation on $A$ and $τ n : A → Z ( A )$ (where $Z ( A )$ is the center of $A$ ) is an $R$ -linear map vanishing at every second commutator $[ [ x , y ] , z ]$ with xy=0 (resp. $x y = p$ ).

Keywords: triangular algebra, Lie higher derivation, Lie triple higher derivation