I think I'm missing something important about tensor products as I look over this question from an old qualifying exam:
Let $R$ be a principal ideal domain and let $A$ and $B$ be finitely generated $R$-modules. Show that if $a\in A$ and $b\in B$ are not torsion elements, then $a\otimes b \neq 0$ in $A\otimes_R B$.
I tried using the definition of tensor products, but did not have any luck. Any suggestions to solve this problem would be appreciated!