Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules

Question

Let $R$ be a commutative Noetherian ring (with unity) and $M,N,P$ be finitely generated projective modules over $R$ such that for some $n\ge 1$, we have $M\otimes_R N \cong M \otimes_R P \cong R^n$. Then is it true that $N \cong P$ ?

Answer 1

In general this is false. For a simple example, take $R=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$, the co-ordinate ring of the real sphere and let $P$ the tangent bundle, given by the presentation, $0\to R\stackrel{(x,y,z)}{\longrightarrow} R^3\to P\to 0$. Then, it is well known that $P$ is not free, but $P\otimes P=P\otimes R^2=R^4$.