# Projective modules are stably isomorphic if localization at a monic polynomial is isomorphic

Let $R$ be a ring and let $P,Q$ be finitely generated projective $R[x]$-modules. Let further $f\in R[x]$ be a monic polynomial. If $P_f\cong Q_f$ (localization on $f$) then $P$ and $Q$ are stably isomorphic, which means that $P\oplus R[x]^n\cong Q\oplus R[x]^n$ for an $n\in\mathbb{N}$.

I can show (using Schanuel's Lemma) that $P$ and $Q$ are stably isomorphic if $P/Q$ is $R$-projective but I don't know if this is the right way.

As you see I am stuck so I am greatful for any hint.

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