Let $M$ be a flat $R$-module. Then the following are equivalent:
1) for every $R$-module $N$ we have $M\otimes_R N\neq0$
2) for every maximal ideal $m$ of $R$ we have $M\neq mM$
I did 1 implies 2. I'm having some problem in 2 implies 1: if $M\otimes_R N=0$ then also $M/mM\otimes_{R/m} N/mN=0$ and so (being vector spaces) or $M=mM$ and we have a contradiction or $N=mN$ but I don't know how to continue, could you help me please?