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