Products of commutators in matrix rings

Journal article, 2025

Let R be a ring and let ≥2. We discuss the question of whether every element in the matrix ring Mn(R) is a product of (additive) commutators [x,y]=xy-yx, for x,y ∈ Mn(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 Mn(R) is a product of three commutators. Further, if R is a division ring with infinite center, then every element in Mn(R) is a product of two commutators. If R is a field and a ∈ Mn(R), then every element in Mn(R) is a sum of elements of the form [a,x] [a,y] with x,y ∈ Mn(R) if and only if the degree of the minimal polynomial of a is greater than 2.