Ideals as generalized prime ideal factorization of submodules

TL;DR

This paper studies the generalized prime ideal factorization of submodules in modules over a ring, establishing conditions under which a given product of prime ideals corresponds to a submodule's factorization.

Contribution

It introduces conditions for the existence of submodules with a specified generalized prime ideal factorization in modules.

Findings

Identifies when a product of prime ideals can be realized as a submodule's factorization.

Provides necessary and sufficient conditions for such factorizations.

Extends the theory of prime ideal factorizations in module theory.

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.