I want to find conditions on which $a \otimes b =0 $ implies $a=0 $ or $b=0$.
My thought is followings.
Let $a\otimes b =0$. $R$ be an integral domain. If $M$ and $N$ are free $R$-modules, $a$ and $ b$ are contained in some finitely generated free submodules $M'$ and $N'$ respectively. For simplicity, let $M'\cong R^m$ and $N' \cong R^n$. If $a= \sum_i^m a_i e_i $ and $b= \sum_j^n b_j f_j$, then $a\otimes b = \sum_{i,j}^{m,n}a_i b_j (e_i \otimes f_j)$. Since $R^m \otimes R^n \cong R^{mn}$, $a_i b_j =0$ for all $i,j$. It follows that $a=0$ or $b=0$.
My questions are as follows.
Is my approach correct?
Is there a better condition?