  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.
