Serial factorizations of right ideals

TL;DR

This paper generalizes the concept of ideal factorizations from Dedekind domains to arbitrary rings, focusing on serial factorizations where modules are uniserial, and explores their uniqueness and connections to other algebraic structures.

Contribution

It introduces and studies serial factorizations of right ideals in arbitrary rings, extending known properties from Dedekind domains and establishing their uniqueness and related algebraic connections.

Findings

Serial factorizations are unique up to order when they exist.

Modules from these factorizations are uniserial.

Connections with $h$-local Prüfer domains and semirigid GCD domains.

Abstract

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal $A$ of an arbitrary ring $R$ as a product of proper right ideals $A_1,\dots,A_n$ with all the modules $R/A_i$ uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of $h$-local Pr\"ufer domains and that of semirigid commutative GCD domains.