PreprintCC BY 4.0arXiv
Unipotence of a double commutator with transvections
Abstract
Let R be a nonzero commutative ring with identity, n≥3, and σ∈ GLn(R). We prove that there exist nonidentity transvections τ1,τ2∈ GLn(R) such that [[σ,τ1],τ2]=In. 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.
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BibTeX
@misc{chinyere2026unipotence,
title = {{Unipotence of a double commutator with transvections}},
author = {Ihechukwu Chinyere},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17006},
eprint = {2609.17006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unipotence of a double commutator with transvections
AU - Ihechukwu Chinyere
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.17006
UR - https://arxiv.org/abs/2609.17006
AB - Let R be a nonzero commutative ring with identity, n≥3, and σ∈ GL_n(R). We prove that there exist nonidentity transvections τ_1,τ_2∈ GL_n(R) such that [[σ,τ_1],τ_2]=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.
ER -