High and Low Formulas in Modules

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. This paper grew out of an attempt to write an introduction to [MRo], and eventually outgrew those confines by becoming an independent paper on two dichotomies in the lattice of unary pp formulas (over any associative ring R with 1). Call a functor from R-Mod to Ab high if it acts as the forgetful functor on all injectives, and low if it vanishes on all flats. Applied to unary positive primitive (henceforth pp) formulas, we obtain high and low formulas with the extra connection that the elementary dual of a high formula (for R-Mod) is a low formula for Mod-R, and v.v. (From now on, unless otherwise specified, formula means unary pp formula.) The first dichotomy is this: every (unary pp) formula that is not high is bounded in the sense that it be annihilated, uniformly in every module, by a single nonzero scalar (and v.v.). No formula is both, high and bounded. Elementary duality yields a second dichotomy: every formula that is not low is cobounded in the sense that one single nonzero scalar sends every R-module into the pp subgroup defined by that formula. And no formula is both, low and cobounded. Date: November 4, 2021. 2020 Mathematics Subject Classification. 16D40, 16D50, 03C60.

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