Zero product and zero Jordan product determined Munn algebras

Abstract

Let $\mathfrak{M}(\mathbb{D}, m, n, P)$ be the ring of all $m \times n$ matrices over a division ring $\mathbb{D}$, with the product given by $A \bullet B=A P B$, where $P$ is a fixed $n \times m$ matrix over $\mathbb{D}$. When $2\leq m, n <\infty$ and $\operatorname{rank} P \geq 2$, we demonstrate that every element in $\mathcal{A}=\mathfrak{M}(\mathbb{D}, m, n, P)$ is a sum of finite products of pairs of commutators. We also estimate the minimal number $N$ such that $\mathcal{A}= \sum^N [\mathcal{A}, \mathcal{A}][\mathcal{A}, \mathcal{A}]$. Furthermore, if $\operatorname{char}\mathbb{D}\neq 2$, we prove that $\mathfrak{M}(\mathbb{D}, m, n, P)$ is additively spanned by Jordan products of idempotents. For a field $\mathbb{F}$ with $\operatorname{char}\mathbb{F}\neq 2, 3$, we show that the Munn algebra $\mathfrak{M}(\mathbb{F}, m, n, P)$ is zero product determined and zero Jordan product determined.