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We prove that if R is a principal ideal ring and A\in\M_n(R) is a matrix withtrace 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 andMuckenhoupt, as well as that over Z due to Laffey and Reams, and as aby-product we obtain new simplified proofs of these results. We also establisha normal form for similarity classes of matrices over PIDs, generalising aresult of Laffey and Reams. This normal form is a main ingredient in the proofof the result on commutators.

Similarity and commutators of matrices over principal ideal rings