tensoring with 0

Question

I'm looking for a simple reason why in $R$-modules

$$0 \otimes M=0 $$

I understand the tensor product $M\otimes N$ as $F(M\times N)/K$

Answer 1

Tensor properties

$(m_1+m_2\otimes n)=(m_1\otimes n)+(m_2\otimes n)$

Hence $M\otimes 0=(M\otimes 0+0)=(M\otimes 0 )+ (M\otimes 0)$ cancel and we are done

Answer 2

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 $r=0$.