Zero product and zero Jordan product determined Munn algebras

TL;DR

This paper investigates the algebraic structure of Munn algebras, showing they are generated by products of commutators and Jordan products of idempotents, and characterizes when they are zero product determined.

Contribution

It establishes new results on the generation and structure of Munn algebras, including conditions for being zero product determined and expressibility via commutators and Jordan products.

Findings

Every element is a sum of finite products of commutators.

Estimated the minimal number of such products needed.

Proved Munn algebras are additively spanned by Jordan products of idempotents under certain conditions.

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.