On trace zero matrices and commutators

Given any commutative ring R, a commutator of two n x n matrices over R has trace 0. In this paper, we study the converse: whether every n x n trace 0 matrix is a commutator. We show that if R is a regular ring with large enough Krull dimension relative to n, then there exists a n x 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 x n trace 0 non -commutator for any n >= 2.(c) 2022 Elsevier Inc. All rights reserved.