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arXiv · 2310.07013

On rational functional identities involving inverses on matrix rings

Yingyu Luo·Qian Chen·Yu Wang
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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.

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Yingyu Luo, Qian Chen, Yu Wang. 2023-09-04. On rational functional identities involving inverses on matrix rings. https://arxiv.org/abs/2310.07013

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