arXiv subjects

Philipp Rothmaler

Publications and source records attributed to Philipp Rothmaler.

9 recordsLinked to original sources

arXiv2025

Bass modules and embeddings into free modules

We show that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15).

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arXiv2024

Free algebras, universal models and Bass modules

We investigate the question of when free structures of infinite rank (in a variety) possess model-theoretic properties like categoricity in higher power, saturation, or universality. Concentrating on left $R$-modules we show, among other things, that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a "stronger" equivalent condition: $R^{(\kappa)}$ purely (equivalently, elementarily) embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. In addition, we extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.

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arXiv2022

Mittag-Leffler modules and definable subcategories. II

In this note I take the opportunity to correct the last statement of Part I of same title and continue the study of uniform purity of epimorphisms in order to derive the main result, which states that--provided $R_R\in \langle\cal K\rangle$, equivalently, $\langle \cal L\rangle$ (the definable subcategory generated by $\cal L$) contains all absolutely pure left modules--every countably generated $\cal K$-Mittag-Leffler module in $\langle \cal L\rangle$ is a direct summand of a $\langle \cal L\rangle$-preenvelope of a union of an $\cal L$-pure $\omega$-chain of finitely presented modules. In conclusion I present a number of examples that starts with and grew out of the study of $\cal L$-purity (of monomorphisms in $\Bbb{Z}$-Mod) for $\cal L$, the definable subcategory of divisible abelian groups. Rings that get particular attention in this are RD-rings, Warfield rings and (the newly introduced) high rings.

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arXiv2021

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.

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arXiv2020

Strict Mittag-Leffler modules and purely generated classes

We study versions of strict Mittag-Leffler modules relativized to a class $\cK$ (of modules), that is, \emph{strict} versions (in the technical sense of Raynaud and Gruson) of $\cK$-Mittag-Leffler modules, as investigated in the preceding paper, {\em Mittag-Leffler modules and definable subcategories}, in this very series (as well as the arXiv).

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arXiv2020

Mittag Leffler modules and definable subcategories

We study (relative) $\mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $\cal K$-Mittag-Leffler modules.

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arXiv2020

Remarks on theories of free algebras and modules

We ask some questions and make some observations about the (complete) theory T (infinity, V) of free algebras in V on infinitely many generators, where V is a variety in the sense of universal algebra. We focus on the case T(infinity, R) where V is the variety of R-modules (R a ring). Building on work in Kucera-Pillay we characterize when all models of T(infinity, R) are free, projective, flat, as well as when T(infinity,R) is categorical in a higher power.

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arXiv2019

Implications of positive formulas in modules (RIMS)

In this survey the role of implications of positive formulas -- finitary and infinitary -- is dicussed, in general and in module categories, where they seem of particular importance. A list of algebraic examples is given, some old, some rather new, and properties are derived from the particular shape of implications involved.

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arXiv2013

Torsion-free, divisible, and Mittag-Leffler modules

We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of absolutely pure modules. A final goal is to describe the K-Mittag-Leffler abelian groups as those that are, modulo their torsion part, aleph_1-free, Cor.6.12. Several more general results of independent interest are derived on the way. In particular, every flat K-Mittag-Leffler module (for K as before) is Mittag-Leffler, Thm.3.9. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open, Quest.4.6.

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