tensoring with 0 I'm looking for a simple reason why in $R$-modules
$$0 \otimes M=0 $$
I understand the tensor product $M\otimes N$ as $F(M\times N)/K$

By construction of the tensor product we have the rule
$$
(rv) \otimes w = r(v \otimes w)
$$
for all $r\in R$, $v\in M$, $w\in N$. Now take $r=0$.
