What are the main cases in which we can say that $a \otimes b \neq 0 \in A \otimes B$, where $A$ and $B$ are $R$ modules?
It works for nonzero elements in free modules over an integral domain.
Additional Question: What can we say about when all tensors are elementary? It is true when one of the factors is cyclic...is that the only reliable principle?
It is necessary and sufficient there exists a bilinear map $A\times B\to M$ for some other module $M$ where $a\otimes b$ has nonzero image.