Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$.
Can we characterize all Noetherian rings $R$ such that for every $R$-module $M$, the $R$-module $\prod_{\mathfrak p \in Spec (R)} M_{\mathfrak p}$ satisfies property $\mathcal P$ ?