Year: 2026
Type: preprint
Abstract: 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 $σ$.
Source: arXiv (Cornell University)
Author: Chinyere, Ihechukwu
Institution: University of Pretoria
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213418626