2012/11/29 by Alexander Stasinski, Stasinski, Alexander
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.1211.6872
23 pages; minor corrections
arxiv created 2013/02/24 · arxiv updated 2013/02/26
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-YX 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.