Additive commutators and trace over a PID

Every matrix with trace zero over a PID is a commutator, according to the MR review of

Rosset, Myriam(IL-BILN); Rosset, Shmuel(IL-TLAV) Elements of trace zero that are not commutators. Comm. Algebra 28 (2000), no. 6, 3059--3072.

From the Math Review:

Although Shoda's method fails when $C$ is a PID, the authors do prove the result in this case, and give counterexamples for $C$ of dimension $\ge 2$.

However, I just took a look at the paper, and as far as I can see the authors only claim the result for 2x2 matrices!

Can anyone resolve this conundrum?

Here (see the very last paragraph) it is stated that every matrix with trace zero over a PID is a commutator. However, I can't come up with a proof right away; the only proof for matrices over a field that I remember (due to Albert?) does not immediately generalize.

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.