can tensor product of two free non zero module over commutative ring with unity be zero?
And can tensor product of two non zero vector spaces be zero space?
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Let $U$ and $V$ be free modules over a nontrivial commutative ring $R$ and let $\alpha:U\to R$ and $\beta:V\to R$ be two $R$-linear maps which have $1$ in their image; such things are easily seen to exist using freeness. Then using the properties of tensor products you can show that there is a morphism of abeelian groups $f:U\otimes_RV\to R$ such that for each $u\in U$ and each $v\in V$ we have $f(u\otimes v)=\alpha(u)\beta(v)$.
Now the hypothesis on $\alpha$ and $\beta$ tells us there exist $u_0\in U$ and $v_0\in V$ with $\alpha(u_0)=1$ and $\beta(v_0)=1$, and then $f(u_0\otimes v_0)=1$. This implies, of course, that $u_0\otimes v_0$ is a nonzero element of $U\otimes_RV$.