Product of prime ideals as factorization of submodules

TL;DR

The paper investigates when a product of prime powers p1^{r1}...pn^{rn} can be realized as the generalized prime ideal factorization P_M(N) of a submodule N in a finitely generated module M over a Noetherian ring. It introduces regular prime extension filtrations (RPE) and establishes multiplicativity P_M(N) = P_M(K)P_K(N) along RPEs, connecting global factorizations to local data and direct sums. A key result is that a power p^r can occur only if p^r ≠ p^{r-1} and Ass(R/p^r) = {p}, and that P_R(p1^{r1} ⊕ ... ⊕ pn^{rn}) = p1^{r1}...pn^{rn} under these conditions; more generally, a product is realizable iff each factor is a GPIF of some submodule. Consequently, realizability reduces to verifying local realizability, yielding a necessary-and-sufficient criterion for the existence of M and N with P_M(N) equal to the given product.

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.

Theorems & Definitions (27)

• Lemma 1.1
• Lemma 1.2
• Example 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Corollary 2.5