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>.