Products of commutators in matrix rings

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

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 $2$.

• Jezik: Angleški jezik
• Ključne besede: commutators, matrix ring, division ring, Bass stable rank, L'vov–Kaplansky conjecture
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.06.2025
• Leto izida: 2025
• Št. strani: Str. 512-529
• Številčenje: Vol. 68, iss. 2
• PID: 20.500.12556/RUL-168780
• UDK: 512
• ISSN pri članku: 0008-4395
• DOI: 10.4153/S0008439524000523
• COBISS.SI-ID: 222178051
• Datum objave v RUL: 24.04.2025

Gradivo je del revije

• Naslov: Canadian mathematical bulletin
• Skrajšan naslov: Can. math. bull.
• Založnik: Cambridge University Press, Canadian Mathematical Society
• ISSN: 0008-4395
• COBISS.SI-ID: 25188608

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.