  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.
