{"title":"On Kursov's theorem for matrices over division rings","authors":"Truong Huu Dung , Tran Nam Son","doi":"10.1016/j.laa.2024.10.018","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>D</em> be a division ring with center <em>F</em> and multiplicative group <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>, where each element of the commutator subgroup of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> can be expressed as a product of at most <em>s</em> commutators. A known theorem of Kursov states that if <em>D</em> is finite-dimensional over <em>F</em>, then every element of the commutator subgroup of the general linear group over <em>D</em> can be expressed as a product of at most <span><math><mi>s</mi><mo>+</mo><mn>1</mn></math></span> commutators. We show that this result remains valid when <em>F</em> has a sufficiently large number of elements, without requiring <em>D</em> to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 218-230"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003987","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let D be a division ring with center F and multiplicative group , where each element of the commutator subgroup of can be expressed as a product of at most s commutators. A known theorem of Kursov states that if D is finite-dimensional over F, then every element of the commutator subgroup of the general linear group over D can be expressed as a product of at most commutators. We show that this result remains valid when F has a sufficiently large number of elements, without requiring D to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.
设 D 是一个中心为 F 的分环和乘法群 D×,其中 D× 的换元子群的每个元素都可以表示为最多 s 个换元的乘积。库尔索夫的一个已知定理指出,如果 D 是 F 上的有限维,那么 D 上一般线性群的换元子群的每个元素都可以表示为最多 s+1 个换元的乘积。我们证明,当 F 有足够多的元素时,这一结果仍然有效,而不需要 D 是有限维的。我们的方法不仅改进了最近关于除法环上矩阵分解的结果,而且还提供了对任意数组上矩阵的恩格尔词映射的研究。
期刊介绍:
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.