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 $A\bullet B=APB$, where $P$ is a fixed $n\times m$ matrix over $\mathbb{D}$. When $2\le m,n<\infty$ and $\mathrm{rank}P\ge 2$, 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 $\mathrm{char}\mathbb{D}\ne 2$, we prove that $\mathfrak{M}(\mathbb{D},m,n,P)$ is additively spanned by Jordan products of idempotents. For a field $\mathbb{F}$ with $\mathrm{char}\mathbb{F}\ne 2,3$, we show that the Munn algebra $\mathfrak{M}(\mathbb{F},m,n,P)$ is zero product determined and zero Jordan product determined.