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EDIT: Not an example because I missed the Noetherian hypothesis. Left here for posterity.

Let R be the ring of continous functions on the 5-sphere. Swan's theorem implies that finitely generated projective modules over this ring are the same thing as vector bundles on $S^5$. The tangent bundle of the 5-sphere is not trivial (it is not parallelizable, which is classical). However, the limiting homotopy group $\varinjlim \pi_5 BO(n)$ classifying stable isomorphism classes of vector bundles is trivial, and so every vector bundle on $S^5$ is stably trivial.
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Let R be the ring of continous functions on the 5-sphere. Swan's theorem implies that finitely generated projective modules over this ring are the same thing as vector bundles on $S^5$. The tangent bundle of the 5-sphere is not trivial (it is not parallelizable, which is classical). However, the limiting homotopy group $\varinjlim \pi_5 BO(n)$ classifying stable isomorphism classes of vector bundles is trivial, and so every vector bundle on $S^5$ is stably trivial.