TITLE: Additive commutators and trace over a PID
QUESTION [11 upvotes]: I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{M}_n(R)$, $A\neq BC-CB$).
I know that such a PID (if it can be found) cannot be a field, or $\mathbb{Z}$.

REPLY [3 votes]: I'm a bit (?) late on that one, but the theorem for general PIDs has been proven after this question was asked.
The reference is (see here for the ArXiv version):

Stasinski, Alexander, Similarity and commutators of matrices over principal ideal rings, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2333–2354.