Certain functional identities on matrix rings

Abstract

Let $D$ be a noncommutative division ring and let $R=M_m(D)$ with $m>1$. We characterize additive mappings $f,$ $g:R\to R$ satisfying the identity $f(X)=X^{n}g(X^{-1})$ for every invertible element $X$ in $R$, where $n$ is a nonnegative integer. We show that the solutions are precisely given by $f=g$ and $f(X)=Xf(I)$ for all $X$ in $R$. Moreover, if $n\ne 2$, both $f$ and $g$ are identically zero.