AI 中文总结
本文针对非交换除环D上的矩阵环R = Mₘ(D)，刻画满足f(X) = Xⁿg(X⁻¹)的加法映射f、g，明确其解的形式，且n≠2时f、g均为零映射。
AI 中文摘要
设D为非交换除环，令R = Mₘ(D)且m > 1。本文刻画满足恒等式f(X) = Xⁿg(X⁻¹)的加法映射f、g:R→R，其中n为非负整数，X为R中任意可逆元。研究表明，解恰好为f = g，且对所有X∈R，f(X) = Xf(I)；此外，若n≠2，则f和g均为零映射。
英文摘要
Let $D$ be a noncommutative division ring and let $R = M_{m}(D)$ with $m > 1$. We characterize additive mappings $f,$ $g:R\rightarrow R$ 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 $n \neq2$, both $f$ and $g$ are identically zero.
