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In our recent work, we introduced a generalization of the prime ideal factorization in Dedekind domains for submodules of finitely generated modules over Noetherian rings. In this article, we find conditions for the intersection of two submodules to have the same factorization as the submodules. We also find the relation between the factorizations of a submodule $N$ in an $R$-module $M$ and the ideal $\mathrm{Ann}(M/N)$ in the ring $R$ and give a condition for their equality.

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.

For a proper submodule $N$ of a finitely generated module $M$ over a Noetherian ring, the product of prime ideals which occur in a regular prime extension filtration of $M$ over $N$ is defined as its generalized prime ideal factorization in $M$. In this article, we find conditions for a product of prime ideals to be the generalized prime ideal factorization of a submodule of some module. We show that a power of a prime ideal occurs in a generalized prime ideal factorization only if it is not equal to its lesser powers. Also, we show that ${{\mathfrak{p}}_1}^{r_1} \cdots {{\mathfrak{p}}_{n}}^{r_{n}}$ is a generalized prime ideal factorization if and only if for each $1 \leq i \leq n$, ${{\mathfrak{p}}_i}^{r_i}$ is the generalized prime ideal factorization of some submodule of a module.