Similarity and commutators of matrices over principal ideal rings

Stasinski, Alexander

doi:10.1090/tran/6402

Similarity and commutators of matrices over principal ideal rings
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Trans. Amer. Math. Soc. 368 (2016), 2333-2354 Request permission

Abstract:

We prove that if $R$ is a principal ideal ring and $A\in \mathrm {M}_{n}(R)$ is a matrix with trace zero, then $A$ is a commutator, that is, $A=XY-YX$ for some $X,Y\in \mathrm {M}_{n}(R)$. This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over $\mathbb {Z}$ 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.

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