Division rings with power commuting semi-linear additive maps

Hereafter, $R$ denotes a noncommutative division ring with centre $Z$, and $f\colon R\to R$ is a semi-linear additive map of $R$ (in the sense given by N. Jacobson, or a more general condition given in the Introduction). In this article, we show that if $f$ is power commuting, that is, (i) there is a positive integer $m$ such that $[f(x),x^m]=0$, all $x\in R$, then $f$ is, in fact, commuting, that is, $[f(x),x]=0$, all $x\in R$. More generally, suppose that (ii) for a fixed pair of positive integers $m$ and $n$, $[f(x),x^m]_n=0$, all $x\in R$. Again, we will show that $f$ is commuting. Now, a doubly more liberal version of the latter condition is Condition (C), which asserts that for each $x$ in $R$, $[f(x),x^{m(x)}]_{n(x)}=0$, where $m(x)$ and $n(x)$ are both positive integers depending on $x$. Unless we are ready to condition appropriately the carrier $R$, the status of Condition (C) remains totally unknown. Granted $R$ is algebraic over $Z$, in particular if $R$ is finite dimensional over $Z$, we show here that if $f$ is an endomorphism or anti-endomorphism of $R$, then from Condition (C) follows again that $f$ is commuting.