2021/11/08 by Makoto Suwama, Suwama, Makoto
Computer Science · Mathematics · #13A70 #16S50 (Primary) 15B33 #52C17 (Secondary) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2111.04884
openalex publication_date 2021/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given any commutative ring R, a commutator of two n× n matrices over R has trace 0. In this paper, we study the converse: whether every n × n trace 0 matrix is a commutator. We show that if R is a Bézout domain with algebraically closed quotient field, then every n× 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× 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× n trace 0 non-commutator for any n≥ 2.