Let $R$ be a commutative unital ring and let $A$ and $B$ be $R$-modules. For any $b\!\in\!B$ the set $$\{a\!\in\!A;\, a\!\otimes\!b\!=\!0\text{ in }A\!\otimes_R\!B\}$$ is a submodule of $A$ that contains $\mathrm{Ann}_R(b)\cdot A$. Is it equal to that product of ideal and $A$?
If not, please provide interesting counterexamples.
No: As a simple example, consider $A:={\mathbb Z}/2{\mathbb Z}$, $B:={\mathbb Q}/{\mathbb Z}$ and their vanishing tensor product ${\mathbb Z}/2{\mathbb Z}\otimes_{\mathbb Z} {\mathbb Q}/{\mathbb Z}$. Taking $b=\frac{1}{2}$ you have $\text{Ann}_{\mathbb Z}(b)=2{\mathbb Z}$, and $2{\mathbb Z}\cdot A=0$. But $\{a\in A\ |\ a\otimes b=0\}=A$ as $A\otimes_{\mathbb Z}B=0$.