Products of commutators in matrix rings

Brešar, Matej (author), Gardella, Eusebio (author), Thiel, Hannes (author)

Let $R$ be a ring and let $n \ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x, y] = xy−yx$, for $x,y \in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a \in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a, x][a, y]$ with $x, y \in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$.

• Language: English
• Keywords: commutators, matrix ring, division ring, Bass stable rank, L'vov–Kaplansky conjecture
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.06.2025
• Year: 2025
• Number of pages: Str. 512-529
• Numbering: Vol. 68, iss. 2
• PID: 20.500.12556/RUL-168780
• UDC: 512
• ISSN on article: 0008-4395
• DOI: 10.4153/S0008439524000523
• COBISS.SI-ID: 222178051
• Publication date in RUL: 24.04.2025

Record is a part of a journal

• Title: Canadian mathematical bulletin
• Shortened title: Can. math. bull.
• Publisher: Cambridge University Press, Canadian Mathematical Society
• ISSN: 0008-4395
• COBISS.SI-ID: 25188608

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.