# Article

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Keywords:
Admissible Lie Ideals; triangular algebra; generalized higher derivation; generalized Jordan higher derivation; generalized Jordan triple higher derivation
Summary:
Let $\mathfrak {A} = \begin {pmatrix}\mathcal {A} & \mathcal {M}\\ &\mathcal {B} \end {pmatrix}$ be the triangular algebra consisting of unital algebras $\mathcal {A}$ and $\mathcal {B}$ over a commutative ring $R$ with identity $1$ and $\mathcal {M}$ be a unital $\mathcal {(A, B)}$-bimodule. An additive subgroup $\mathfrak { L }$ of $\mathfrak { A }$ is said to be a Lie ideal of $\mathfrak {A}$ if $[\mathfrak {L},\mathfrak {A}]\subseteq \mathfrak {L}$. A non-central square closed Lie ideal $\mathfrak { L }$ of $\mathfrak { A }$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $\mathfrak {A}$, every generalized Jordan triple higher derivation of $\mathfrak {L}$ into $\mathfrak {A}$ is a generalized higher derivation of $\mathfrak {L}$ into $\mathfrak { A }$.
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