Multiplicative Lie-type derivations on alternative rings

Abstract

Let $\R$ be an alternative ring containing a nontrivial idempotent and $\D$ be a multiplicative Lie-type derivation from $\R$ into itself. Under certain assumptions on $\R$, 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 $\R$ 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 $\R$ and $\D$, that $\D=\delta+\tau$, where $\delta$ is a derivation and $\tau\colon\R\longrightarrow{\mathcal Z}(\R)$ such that $\tau(p_n(a_1,\ldots,a_n))=0$ for all $a_1,\ldots,a_n\in\R$.