Similarity and commutators of matrices over principal ideal rings

Alexander Stasinski
2015 Transactions of the American Mathematical Society  
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more » ... . Please consult the full DRO policy for further details. Abstract. We prove that if R is a principal ideal ring and A ∈ Mn(R) is a matrix with trace zero, then A is a commutator, that is, A = XY − Y X for some X, Y ∈ Mn(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.
doi:10.1090/tran/6402 fatcat:mbz3z5sat5ggrhodwxtrsqqh6u