I've read some materials about tensor product,but there are things still confusing to me. Briefly speaking, I still wonder about the elements in a tensor product.
(1) How can we prove that some elements in tensor product are not equal?
For example, how to prove $3 \otimes \overline{2} \ne 0$ in $\mathbb{Z}[x] \otimes \mathbb{Z}/5\mathbb{Z}$? How about more complicated elements? Such as $3x \otimes \overline{4} + x^5 \otimes \overline{3} \ne 0$ although I believe it's not zero because I can't reduce it to $0$.
(2)How can we proof the right-exactness of tensor product directly?
I've been tried to proof the right-exactness of tensor product,but always get confused when dealing with specific elements of them. Is there a proof without using the "adjoint with $\rm Hom$" method?