与平延的双换位子的幂单性

中文摘要 AI 辅助

我们证明如下命题：对每个 $\sigma \in GL_n(\mathbb{R})$（其中 $n \geq 3$），存在平延 $\tau_1, \tau_2 \in GL_n(\mathbb{R})$，使得 $\left(\big[[\sigma, \tau_1], \tau_2\big]-I\right)^2=0$。特别地，$\big[[\sigma, \tau_1], \tau_2\big]$ 是幂单的。证明依赖于实数自同态存在一维或二维不变子空间这一事实。

英文摘要

Let $R$ be a nonzero commutative ring with identity, $n\geq3$, and $σ\in GL_n(R)$. We prove that there exist nonidentity transvections $τ_1,τ_2\in GL_n(R)$ such that $\big[[σ,τ_1],τ_2\big]=I_n$. Hence, Kourovka Notebook Problem 10.46 has a stronger affirmative answer over commutative rings with identity. The proof uses rank-one operators and the existence of a nonzero functional annihilating a standard basis vector and its image under $σ$.

发表机构

• University of Pretoria（比勒陀利亚大学）

机构由 AI 辅助整理，请以论文原文为准。