MMN-1851
Traces of permuting generalized $n$-derivations of rings
Mohammad Ashraf; Almas Khan; Malik Rashid Jamal;Abstract
Let $n\geq 1$ be a fixed positive integer and $R$ be a ring.
A permuting $n$-additive map $\Omega:R^n\to R$ is known to be
permuting generalized $n$-derivation if there exists a permuting $n$-derivation $\Delta:R^n\to R$
such that $\Omega(x_1,x_2,\cdots, x_ix_i^{'},\cdots, x_n)=\Omega(x_1,x_2,\cdots, x_i,\cdots, x_n)x_i^{'}+
x_i\Delta(x_1,x_2,\cdots, x_i^{'},\cdots, x_n)$
holds for all $x_i ,x_i^{'} \in R$.
A mapping $\delta:R \to R$ defined by
$\delta(x)=\Delta(x,x,\cdots,x)$ for all $x\in R$ is said to be the trace of $\Delta$.
The trace $\omega$ of $\Omega$ can be defined in the similar way.
The main result of the present paper states that if $R$ is a $(n+1)!$-torsion free semi-prime ring which admits a permuting $n$-derivation
$\Delta$ such that the trace $\delta$ of $\Delta$ satisfies $[[\delta(x),x],x]\in Z(R)$ for all $x\in R,$ then
$\delta$ is commuting on $R$. Besides other related results it is also shown that in a $n!$-torsion free prime ring if the trace $\omega$
of a permuting generalized $n$-derivation $\Omega$ is centralizing on $R,$ then $\omega$ is commuting on $R$.
Vol. 19.0 (2018), No. 2.0, pp. 731-740