  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.
