Certain functional identities on division rings of characteristic two (Q6588196)

The immediate context of this paper is [\textit{N. A. Dar} and \textit{W. Jing}, Quaest. Math. 46, No. 5, 927--937 (2023; Zbl 1532.16020); \textit{T.-K. Lee} and \textit{J.-H. Lin}, J. Algebra 647, 492--514 (2024; Zbl 1545.16023)] in which the authors studied non-commutaitve division rings admitting a pair of additive maps \(f,g:D\rightarrow D\), satisfying \(f(x)+x^ng(x^{-1})=0\), for all non-zero~\(x\) and for some fixed positive integer~\(n\). Together, these two papers completely characterized such maps outside of characteristic~2, and the current paper completes the story by analizing the characteristic~2 case. Here is the conclusion: If \(n=2\), then \(f(x)=g(x)=xf(1)\), for all~\(x\); and if \(n\ne 2\), then \(f=g=0\).