Similarity and commutators of matrices over principal ideal rings

TL;DR

This paper proves that matrices with trace zero over principal ideal rings are commutators, extending known results from fields and integers, and introduces a normal form for similarity classes over PIDs.

Contribution

It generalizes the commutator characterization for matrices over principal ideal rings and provides a new normal form for similarity classes over PIDs.

Findings

Matrices with trace zero over principal ideal rings are commutators.

Established a normal form for similarity classes of matrices over PIDs.

Provided simplified proofs of existing results over fields and integers.

Abstract

We prove that if R is a principal ideal ring and A\in\M_n(R) is a matrix with trace zero, then A is a commutator, that is, A=XY-YX for some X,Y\in\M_n(R). This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over Z due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators.