Submodules having the same generalized prime ideal factorization

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 $\mathrm{A}\mathrm{n}\mathrm{n}(M\mathrm{/}N)$ in the ring $R$ and give a condition for their equality.