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$n$-HIGHER DERIVATIONS AND LIE 
$n$-HIGHER DERIVABLE MAPPINGS
Let 
${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring 
${\mathcal{R}}$. To characterise Lie 
$n$-higher derivations on 
${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie 
$n$-higher derivations into the same problems concerning Lie 
$n$-derivations. We prove that: (1) if every Lie 
$n$-derivation on 
${\mathcal{A}}$ is standard, then so is every Lie 
$n$-higher derivation on 
${\mathcal{A}}$; (2) if every linear mapping Lie 
$n$-derivable at several points is a Lie 
$n$-derivation, then so is every sequence 
$\{d_{m}\}$ of linear mappings Lie 
$n$-higher derivable at these points; (3) if every linear mapping Lie 
$n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all 
$(n-1)$th commutators of these points, then every sequence 
$\{d_{m}\}$ of linear mappings Lie 
$n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all 
$(n-1)$th commutators of these points. We also give several applications of these results.
