Ideals as generalized prime ideal factorization of submodules
K. R. Thulasi, T. Duraivel, S. Mangayarcarassy
TL;DR
The paper investigates when a prescribed product of prime ideals can be realized as the generalized prime ideal factorization of a submodule in a finitely generated module over a Noetherian ring. It develops the framework of MPE and RPE filtrations to track primes in ${\mathrm{Ass}}(M/N)$ and defines ${\mathcal{P}}_M(N)$ as the product of these primes along a filtration. It provides explicit necessary and sufficient conditions for realizing products ${\mathfrak{p}}_1^{r_1}\cdots{\mathfrak{p}}_n^{r_n}$ (with distinct ${\mathfrak{p}}_i$) as ${\mathcal{P}}_M(N)$ in terms of $\mathrm{Supp}$ of certain $M$-modules, and extends to repeated primes via colon-ideal criteria, including a criterion for ${\mathcal{P}}_M({\mathfrak{a}}M)={\mathfrak{a}}$ when ${\mathfrak{a}}$ is square-free. These results yield constructive filtrations for realizable factorizations and illuminate limitations where some products cannot occur.
Abstract
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.