Multiplicative Lie-type derivations on alternative rings

Abstract

Let $\mathbb{R}$ be an alternative ring containing a nontrivial idempotent and $\D$ be a multiplicative Lie-type derivation from $\mathbb{R}$ into itself. Under certain assumptions on $\mathbb{R}$, we prove that $\D$ is almost additive. Let $p_n(x_1,x_2,\cdots ,x_n)$ be the $(n-1)$-th commutator defined by $n$ indeterminates $x_1,\cdots ,x_n$. If $\mathbb{R}$ is a unital alternative ring with a nontrivial idempotent and is $\{2,3,n-1,n-3\}$-torsion free, it is shown under certain condition of $\mathbb{R}$ and $\D$, that $\D=δ+τ$, where $\delta$ is a derivation and $\tau :\mathbb{R}⟶\mathcal{Z}(\mathbb{R})$ such that $\tau (p_n(a_1,\dots ,a_n))=0$ for all $a_1,\dots ,a_n\in \mathbb{R}$.

23 pages