 Local tensor nonvanishing criterion

A nonzero commutative ring $R$ has the property that $M{\otimes }_{R}N=0$ forces $M=0$ or $N=0$ exactly when $R$ is local with maximal ideal $\mathfrak{m}$ and $M=\mathfrak{m}M$ forces $M=0$ for every $R$-module. Reduction modulo $\mathfrak{m}$ turns a tensor product into a tensor product of vector spaces.