Let $A$ be a commutative ring with unit $1$ and $M$, $N$ be $A$-modules. Consider tensor product of $M\otimes N$. I wonder when $x\otimes y=0$, what can we say about $x$ and $y$?
I know that if $x=0$ or $y=0$, we get $x\otimes y=0$, or when $x=at$ for some $a\in A,t \in M$ and $ay=0$ then $x\otimes y=0$, too.