For a module $M$ over a commutative ring $R$, let
$\operatorname{Ass}_R (M):=\{\operatorname{ann}_R (m) \mid m\in M$ and $\operatorname{ann}_R(m) \in \operatorname{Spec}(R)\}$.
If $M$ is a finitely generated $R$-module and $P \in \operatorname{Ass}_R(M)$ and $N$ is a flat $R$-module, then is it true that $\operatorname{Ass}_R(N/PN)\subseteq \operatorname{Ass}_R(M \otimes_R N) $ ?
If $R$ is moreover Noetherian, then how to prove that $Ass(M \otimes N)=\cup_{P \in Ass(M)} Ass(N/PN)$ ?