Inner Higher Derivations on Algebras

E. Tafazoli, M. Mirzavaziri

Let $\mathcal{A}$ be an algebra. A sequence $\mathbf{d}=\{d_n\}_{n=0}^\infty$ of linear operators on $\mathcal{A}$ is called a extit{higher derivation} if $d_0$ is the identity mapping on $\mathcal{A}$ and $d_n(xy)=\sum_{k=0}^nd_k(x)d_{n-k}(y)$, for each $n=0,1,2,\ldots$ and $x,y\in\mathcal{A}$. We say that a higher derivation $\mathbf{d}$ is extit{inner} if there is a sequence $\mathbf{a}=\{a_n\}_{n=1}^\infty$ in $\mathcal{A}$ such that inebreak $(n+1)d_{n+1}(x)=\sum_{k=0}^n a_{k+1}d_{n-k}(x)-d_{n-k}(x)a_{k+1}$, for each $n=0,1,2,\ldots$ and $x\in\mathcal{A}$. Giving a characterization for inner higher derivations on a torsion free algebra $\mathcal{A}$, we show that each higher derivation on $\mathcal{A}$ is inner provided that each derivation on $\mathcal{A}$ is inner.