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 $\mathbb{Z}$.