Certain functional identities on division rings of characteristic two

Abstract

Let $D$ be a noncommutative division ring. In a recent paper, Lee and Lin proved that if $\text{char}D\ne 2$, the only solution of additive maps $f,g$ on $D$ satisfying the identity $f(x)=x^{n}g(x^{-1})$ on $D\setminus \{0\}$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer $n$ if $\text{char}D=2$.