Ideals as generalized prime ideal factorization of submodules

For a submodule $N$ of an $R$-module $M$, a unique product of prime ideals in $R$ is assigned, which is called the generalized prime ideal factorization of $N$ in $M$, and denoted as ${\mathcal{P}}_M(N)$. But for a product of prime ideals ${\mathfrak{p}}_1\cdots {\mathfrak{p}}_n$ in $R$ and an $R$-module $M$, there may not exist a submodule $N$ in $M$ with ${\mathcal{P}}_M(N)={\mathfrak{p}}_1\cdots {\mathfrak{p}}_n$. In this article, for an arbitrary product of prime ideals ${\mathfrak{p}}_1\cdots {\mathfrak{p}}_n$ and a module $M$, we find conditions for the existence of submodules in $M$ having ${\mathfrak{p}}_1\cdots {\mathfrak{p}}_n$ as their generalized prime ideal factorization.