Zero product and zero Jordan product determined Munn algebras

Let M(D, m, n, P) be the ring of all m × n matrices over a division ring D, with the product given by A • B=A P B, where P is a fixed n × m matrix over D. When 2≤ m, n <∞ and rank P ≥ 2, we demonstrate that every element in A=M(D, 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 charD≠ 2, we prove that M(D, m, n, P) is additively spanned by Jordan products of idempotents. For a field F with charF≠ 2, 3, we show that the Munn algebra M(F, m, n, P) is zero product determined and zero Jordan product determined.