On rational functional identities involving inverses on matrix rings

TL;DR

This paper characterizes additive maps on matrix rings over division rings, showing that under certain conditions, the only solutions to a specific functional identity involving inverses are trivial zero maps.

Contribution

It generalizes previous results by proving that the only additive solutions to a particular inverse-related functional equation are zero maps in matrix rings over division rings.

Findings

Only zero maps satisfy the functional identity under given conditions

Generalizes prior results by Dar, Jing, Catalano, and Merchán

Applicable to matrix rings over division rings with specified characteristics

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.