Generalized Higher Derivations on Lie Ideals of Triangular Algebras

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 }$