On Trace Zero Matrices and Commutators

TL;DR

This paper investigates conditions under which trace zero matrices over various rings are expressible as commutators, providing positive results for certain rings and counterexamples for others, thus advancing understanding of matrix commutator properties.

Contribution

The paper establishes new conditions where trace zero matrices are commutators over specific rings and constructs counterexamples, extending previous results and exploring ring-theoretic influences.

Findings

Over Bézout domains with algebraically closed quotient fields, all trace zero matrices are commutators.

Counterexamples exist in regular rings with high Krull dimension, where some trace zero matrices are not commutators.

A Noetherian dimension 1 domain admits trace zero matrices that are not commutators for any size n ≥ 2.

Abstract

Given any commutative ring $R$, a commutator of two $n\times n$ matrices over $R$ has trace $0$. In this paper, we study the converse: whether every $n \times n$ trace $0$ matrix is a commutator. We show that if $R$ is a B\'{e}zout domain with algebraically closed quotient field, then every $n\times n$ trace $0$ matrix is a commutator. We also show that if $R$ is a regular ring with large enough Krull dimension relative to $n$, then there exist a $n\times n$ trace $0$ matrix that is not a commutator. This improves on a result of Lissner by increasing the size of the matrix allowed for a fixed $R$. We also give an example of a Noetherian dimension $1$ commutative domain $R$ that admits a $n\times n$ trace $0$ non-commutator for any $n\ge 2$.