If Tensor Product of Two Finitely Generated Modules Over a Local Ring is Zero then one of the Modules is Zero

Question

Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules.

If $M\otimes_R N=0$, then $M=0$ or $N=0$.

The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where $\mathfrak m$ is the unique maximal ideal of $M$, then by Nakayama we would have $M=0$ or $N=0$, for the Jacobson radical of $M$ is nothing but $\mathfrak m$ itself.

I am unable to figure out what to do.

Answer 1

$$M\otimes_RN=0\Rightarrow M\otimes_RN\otimes_RR/m=0\Rightarrow M\otimes_RN/mN=0\Rightarrow (M\otimes_RR/m)\otimes_{R/m}N/mN=0\Rightarrow M/mM\otimes_{R/m}N/mN=0$$