Let $R$ be a Commutative Noetherian ring. Let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism classes of finitely generated $R$-modules subject to the relation : $[M]=[M_1]+[M_2]$ if there is a short exact sequence of $R$-modules $0\to M_1\to M\to M_2\to 0$.
It can be shown that $G_0(R)$ is generated by the classes $[R/P]$ as $P$ runs over all prime ideals of $R$. Now my question is the following:
If $M,N$ are finitely generated $R$-modules such that $[M]=[N]$ in $G_0(R)$, then is it true that there exists short exact sequences of finitely generated $R$-modules $0\to A\to B\to C\to 0$ and $0\to A\to B'\to C\to 0$ such that $M\oplus B\cong N\oplus B'$ ?
Something similar was once pointed out by Steven Landsburg, that works in any exact category: there are exact sequences $0 \to A \to B\to C \to 0$ and $0 \to A' \to B'\to C' \to 0$ such that $ M \oplus B \oplus C \oplus A' \cong N \oplus B' \oplus C' \oplus A$. It's an easy exercise.