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Similarity and commutators of matrices over principal ideal rings


Author: Alexander Stasinski
Journal: Trans. Amer. Math. Soc. 368 (2016), 2333-2354
MSC (2010): Primary 15B36; Secondary 15B33
Published electronically: July 10, 2015
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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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Additional Information

Alexander Stasinski
Affiliation: Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, United Kingdom
Email: alexander.stasinski@durham.ac.uk

DOI: https://doi.org/10.1090/tran/6402
Received by editor(s): February 26, 2013
Received by editor(s) in revised form: April 4, 2013, October 4, 2013, and January 10, 2014
Published electronically: July 10, 2015
Article copyright: © Copyright 2015 American Mathematical Society