Product of prime ideals as factorization of submodules

Abstract

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.