20.500.12556/DiRROS-22101 Products of commutators in matrix rings 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 $2$. commutators matrix ring division ring Bass stable rank L'vov–Kaplansky conjecture true false true Angleški jezik Ni določen Neznano 2025-04-24 09:20:07 2025-04-24 09:20:07 2025-10-24 03:50:32 0000-00-00 00:00:00 2025 0 0 str. 512-529 iss. 2 Vol. 68 Jun. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-06-01 512 0008-4395 10.4153/S0008439524000523 222178051 products_of_commutators_in_matrix_rings.pdf products_of_commutators_in_matrix_rings.pdf 1 7837A45086389740007351A529851B17 50d43eb148429a2557a566c9b8ccd142300978eb139f3152e2edd8bc8e0c8b4c aa37c0c0-20dd-11f0-963a-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=31950 https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/products-of-commutators-in-matrix-rings/10FD7B61EB100163AA3815437915BA66 0 b03c0bdf-20dd-11f0-963a-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=31951 Inštitut za matematiko, fiziko in mehaniko 0 0 0