Grothendieck group of artinian abelian category

Question

Let $\mathscr{C}$ be an artinian, abelian category and $K(\mathscr{C})$ its Grothendieck group. If $[A]=0$ in $K(\mathscr{C})$ for an object $A$, can we then conclude $A=0$? This is true if $\mathscr{C}$ is also noetherian (i.e. finite-length), but I can't seem to show it if $\mathscr{C}$ is only artinian.

Answer 1

There are familiar Noetherian counterexamples. For example, if $\mathcal{C}$ is the category of finitely generated abelian groups, then $[A]=0\in K_0(\mathcal{C})$ for any finite abelian group $A$.

For an Artinian example, just take the opposite category of a Noetherian example.