High and Low Formulas in Modules

TL;DR

This paper introduces a partition of unary pp formulas into four regions, including high formulas, to analyze module structures and applies this to various rings, revealing properties of pure injective modules and their decompositions.

Contribution

It develops a new framework dividing unary pp formulas into four regions, enabling analysis of modules over rings and domains, and characterizes pure injective modules using Ulm length.

Findings

Pure injective modules have Ulm length at most 1.

Pure injective modules over RD domains decompose into injective and reduced parts.

The framework applies to various rings, including Ore domains and the Weyl algebra.

Abstract

A partition of the set of unary pp formulas into four regions is presented, which has a bearing on various structural properties of modules. The machinery developed allows for applications to IF, weakly coherent, nonsingular, and reduced rings, as well as domains, specifically Ore domains. One of the four types of formula are called high. These are used to define Ulm submodules and Ulm length of modules over any associative ring. It is shown that pure injective modules have Ulm length at most 1. As a consequence, pure injective modules over RD domains (in particular, pure injective modules over the first Weyl algebra over a field of characteristic 0) are shown to decompose into a largest injective and a reduced submodule. This study serves as preparation for forthcoming work with A. Martsinkovsky on injective torsion.