# Tag Archives: commutator

## Commutators and the Ore Conjecture

In a talk yesterday by Boris Kunyavski at the University of Ottawa, I learned a little about the Ore conjecture, which in 2010 was proved a theorem in: Liebeck, Martin W.; O'Brien, E. A.; Shalev, Aner; Tiep, Pham Huu. The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008. It's quite a […]

## For (most) PIDs: Trace zero matrices are commutators

Let $R$ be a commutative ring and $M_n(R)$ denote the ring of $n\times n$ matrices with coefficients in $R$. For $X,Y\in M_n(R)$, their commutator $[X,Y]$ is defined by $$[X,Y] := XY – YX.$$ The trace of any matrix is defined as the sum of its diagonal entries. If $X$ and $Y$ are any matrices, what […]