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 $\mathbb{Z}$.



Additive commutators and trace over a PID
It is not difficult to see that Rosset & Rosset's result for $2\times2$ matrices is equivalent to the surjectivity of the bilinear map $(X,Y)\mapsto X\times Y$ (called *vector product* when $A={\mathbb R}^3$) over $A^3$. For this, just search $B$ and $C$ such that $b\_{22}=c\_{22}=0$.


To prove it, let $Z=(a,b,c)\in A^3$ be given. One can choose a primitive vector $X=(x,y,z)$ such that $ax+by+cz=0$. By *primitive*, I mean that $gcd(x,y,z)=1$. Bézout tells that there exist a vector $U=(u,v,w)$ such that $ux+vy+wz=1$. Set $Y=Z\times U$. Then $Z=X\times Y$. 


