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 ≥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?