# Rings and categories with zero Grothendieck group

I am interested in examples of rings (or triangulated categories) that have zero Grothendieck group but are somehow still interesting. More example, for what rings $$R$$ is the category of finitely-generated projective $$R$$-modules have zero Grothendieck group? I recently learned that the category of all modules has zero Grothendieck group: if $$M$$ is any $$R$$-module, then $$M \oplus M^{\oplus \infty} \cong M^{\oplus \infty}$$ and so $$M$$ is zero in the Grothendieck group. I also know that "infinite sum rings" with the property that $$R \oplus R \cong R$$ have vanishing Grothendieck group. I would like some "smaller" examples of rings with vanishing Grothendieck group. Also, such an $$R$$ cannot be commutative ring, since commutative rings have invariant basis property and hence have non-zero Grothendieck group.

It is easy to construct categories that are not split-closed where the Grothendieck group is zero. For example, suppose that some triangulated is generated by an object $$A$$. Then the subcategory generated by $$A \oplus A[1]$$ has zero Grothendieck group because the shift acts like $$-1$$ in the Grothendieck group; however this category is not split-closed since the projective object $$A$$ is not in the category. So I would like an example of a split-closed category that has zero Grothendieck group.

Maybe an easier but less concrete question: what does the vanishing of the Grothendieck group imply? By a note in a paper of Thomason, $$D$$ is zero in the Grothendieck group if there exist $$A,B,C$$ and exact triangles $$A \rightarrow B \oplus D \rightarrow C \rightarrow$$ and $$A \rightarrow B \rightarrow C \rightarrow$$ but this is not very enlightening. Does someone have another interpretation?