Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 3 / a / ii / Solution

Let $k=R/\mathfrak{m}$. Associativity and part i giveIf $M{\otimes }_{R}N=0$, the tensor product on the right is zero. Two nonzero vector spaces over a field have nonzero tensor product, so one factor vanishes. Nakayama lemma then gives $M=0$ or $N=0$.