Division rings with power commuting semi-linear additive maps
Journal
Publicationes Mathematicae
Journal Volume
95
Journal Issue
43832
Pages
187-203
Date Issued
2019
Author(s)
Chacron, M.
Abstract
Hereafter, R denotes a noncommutative division ring with centre Z, and f : R ! 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); xm] = 0, all x 2 R, then f is, in fact, commuting, that is, [f(x); x] = 0, all x 2 R. More generally, suppose that (ii) for a fixed pair of positive integers m and n, [f(x); xm]n = 0, all x 2 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); xm(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.
Type
journal article