On rational functional identities involving inverses on matrix rings

Abstract

Let $n\geq 3$ be an integer. Let $\mathcal{D}$ be a division ring with char$(\mathcal{D})>n$ or char$(\mathcal{D})=0$. Let $\mathcal{R}=M_m(\mathcal{D})$ be a ring of $n\times n$ matrices over $D$, $m\geq 2$. The main theorem in the paper states that the only additive maps $f$ and $g$ satisfying that $f(X)+X^ng(X^{-1})=0$ for all invertible $X\in \mathcal{R}$, are zero maps, which generalizes both a result proved by Dar and Jing and a result proved by Catalano and Merch$\acute{a}$n.