High and Low Formulas in Modules
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Publication:6382023
arXiv2111.01986MaRDI QIDQ6382023
Publication date: 2 November 2021
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.
Model-theoretic algebra (03C60) Injective modules, self-injective associative rings (16D50) Free, projective, and flat modules and ideals in associative algebras (16D40)
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