Products of commutators in matrix ringsBrešar, Matej (Avtor) Gardella, Eusebio (Avtor) Thiel, Hannes (Avtor) commutatorsmatrix ringdivision ringBass stable rankL'vov–Kaplansky conjectureLet $R$ 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 $2$.20252025-04-24 09:20:07Neznano22101UDK: 512ISSN pri članku: 0008-4395DOI: 10.4153/S0008439524000523COBISS_ID: 222178051sl