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<p>Let <i>R</i> be a commutative associative ring with 1, and let <i>G =</i> GL(<i>n</i>,<i>R</i>) be the general linear group of degree <i>n</i> ≥ 3 over <i>R</i>. Further, let <i>I</i> ⊴ <i>R</i> be an ideal of <i>R</i>. In the present note, which is a marginalia to the paper of Alexei Stepanov and the second author (2000), explicit expressions of the elementary transvection <i>gt</i><sub><i>ij</i></sub>(ξ)<i>g</i><sup><i>−</i>1</sup>, where 1 ≤ <i>i</i> ≠ <i>j</i> ≤ <i>n</i>, ξ ∈ <i>I</i> and <i>g</i> ∈ <i>G</i>, are obtained as products of the Stein–Tits–Vaserstein generators of the relative elementary group <i>E</i>(<i>n,R, I</i>)<i>.</i></p>

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Relative Decomposition of Transvections: Explicit Bounds

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • M. A. Buryakov,
  • N. A. Vavilov

摘要

Let R be a commutative associative ring with 1, and let G = GL(n,R) be the general linear group of degree n ≥ 3 over R. Further, let I ⊴ R be an ideal of R. In the present note, which is a marginalia to the paper of Alexei Stepanov and the second author (2000), explicit expressions of the elementary transvection gtij(ξ)g−1, where 1 ≤ i ≠ j ≤ n, ξ ∈ I and g ∈ G, are obtained as products of the Stein–Tits–Vaserstein generators of the relative elementary group E(n,R, I).