Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 101 / 1 / ii / c

The statement is true. Fix an isomorphism $\varphi :M{\otimes }_{R}N\to R^n$, and write the inverse image of the first standard basis vector asDefineSince $\psi ((n_j{)}_j)=1$, this is a split surjection. Tensor its splitting with $M$. The resulting split surjection has the formThus $M$ is a direct summand of a finite free module, so it is a projective module. Symmetry gives the same conclusion for $N$. This is projectivity of factors of a nonzero finite free tensor product.