SIMILARITY AND COMMUTATORS OF MATRICES OVER PRINCIPAL IDEAL RINGS

We prove that if R is a principal ideal ring and A is an element of M-n (R) is a matrix with trace zero, then A is a commutator, that is, A = XY - YX for some X, Y is an element of 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.