2016/07/25 by Alexander Stasinski, Stasinski, Alexander
Computer Science · Mathematics · #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1607.07205
openalex publication_date 2016/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for every trace zero matrix A over a principal ideal ring R, there exist trace zero matrices X,Y over R such that XY-YX=A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one. Shalev has conjectured an analogous statement for group commutators in SLn over p-adic integers. We prove Shalev's conjecture for n=2.