2024/07/01 by Yu, Bo, Luo, Kaijia, Li, Jiankui
#16S50 (Primary) 15A30 (Secondary) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2407.00892
Let \mathfrakM(\mathbbD, m, n, P) be the ring of all m × n matrices over a division ring \mathbbD, with the product given by A \bullet B=A P B, where P is a fixed n × m matrix over \mathbbD. When 2≤ m, n <∞ and rank P ≥ 2, we demonstrate that every element in A=\mathfrakM(\mathbbD, m, n, P) is a sum of finite products of pairs of commutators. We also estimate the minimal number N such that A= ∑N [A, A][A, A]. Furthermore, if char\mathbbD≠ 2, we prove that \mathfrakM(\mathbbD, m, n, P) is additively spanned by Jordan products of idempotents. For a field \mathbbF with char\mathbbF≠ 2, 3, we show that the Munn algebra \mathfrakM(\mathbbF, m, n, P) is zero product determined and zero Jordan product determined.