Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are stably isomorphic, i.e. $M\oplus R^a\cong N\oplus R^a$, then clearly $M_{(p)}\cong N_{(p)}$. I was wondering about the converse. When, if ever, does $M_{(p)}\cong N_{(p)}$ imply $M\oplus R^a\cong N\oplus R^a$?
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