Model Theoretic Algebra With Particular Emphasis on Fields, Rings, Modules

Christian.U Jensen, Christian. U Jensen, Helmt Lenzing

• Publisher: CRC Press
• Year: 2022
• Print ISBN: 9782881247170
• eBook ISBN: 9781351431118
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This volume highlights the links between model theory and algebra. The work contains a definitive account of algebraically compact modules, a topic of central importance for both module and model theory. Using concrete examples, particular emphasis is given to model theoretic concepts, such as axiomizability. Pure mathematicians, especially algebraists, ring theorists, logicians, model theorists and representation theorists, should find this an absorbing and stimulating book.

Chapter 1 Introduction. Ultraproducts. Definitions and examples

DOI: 10.4324/9780203746943-1

The aim of this introductory chapter is to illustrate the power of the most basic principles of model theory (ultraproducts, Ƚos’s principle, Löwenheim-Skolem’s theorem and Keisler-Shelah’s ultrapower theorem) in applying them to classical questions of algebra (Hilbert’s Nullstellensatz, Hilbert’s 17^th problem, Noether-Ostrowski’s irreducibility theorem).

For a detailed treatment of the fundamental facts of model theory we refer the reader to the the books of Bell-Slomson [21], Chang-Keisler [33] or the Handbook of Mathematical Logic [12].

Ultrafilters

1.1 If I is an arbitrary set, we recall that a filter ℱ on I is a family of subsets of I satisfying the following conditions:

1. A ∈ ℱ, A ⊆ B ⊆ I ⇒ B ∈ ℱ,
2. A ∈ ℱ and B ∈ ℱ ⇒ A ∩ B ∈ ℱ,
3. Ø ∉ ℱ.

The filters on I are partially ordered by set inclusion. Filters that are maximal with respect to this ordering are called ultrafilters. It is easy to see that ultrafilters can be characterized in the following way:

1.2 A filter ℱ on I is an ultrafilter if and only if for each subset A of I either A or the complement I – A belongs to ℱ.

For a fixed element α ∈ I the family of all subsets of I containing α is an ultrafilter on I, called the principal ultrafilter generated by a. By Zorn’s lemma any filter on I can be extended to an ultrafilter on I. This, in particular, implies that there exist non-principal ultrafilters on any infinite set I.

Ultraproducts and Ƚos’s principle

1.3 Let (R_α)_α∈I be a family of algebraic structures which may be groups, rings or modules. To fix the attention on something specific let us assume that (R_α)_α∈I is a family of rings. Further, let ℱ be an ultrafilter on I. In the direct (Cartesian) product Π_α∈I R_α we introduce an equivalence relation by setting (r_α) ~ (r’_α) if the set {α∈I|r_α = r′_α} belongs to ℱ. This is expressed by saying that

$\left(r_{\alpha }\right)~\left(r{\text{'}}_{\alpha }\right)$

if and only if r_α = r′_α holds for ℱ-almost all α.

The equivalence class represented by an element (r_α) is denoted by [r_α]. By obvious componentwise addition and multiplication these equivalence classes form a ring, called the ultraproduct of (R_α)_α∈I with respect to ℱ, that is denoted Π_α∈I R_α/ℱ. If R_α= R for all α ∈ I, the ultraproduct is denoted by R^I/ℱ and is called the ultrapower of R with respect to ℱ. In the latter case there is a canonical (diagonal) mapping Δ: R →R^I/ℱ defined by setting Δ(r) = [r_α], where r_α = r for all α ∈ I.

1.4 It is easily verified that Π_α∈I R_α/ℱ is a field if each R_α is a field. It even suffices to assume that R_α is a field for all α ’s belonging to a subset J of I that belongs to ℱ. Conversely, if the ultraproduct Π_α∈I R_α/ℱ is a field, the set {α ∈IR_α is a field } belongs to ℱ. In other words: the ultraproduct Π_α∈I R_α. is a field if and only if R_α is a field for ℱ-almost every α ∈ I.

The property R is a field can be expressed by saying

${\forall }_{x\ne 0}{\exists }_{y}xy=1.$

This is an example of a first order sentence in the language R of rings, that is, a formula in the language of rings, in which every variable is in the scope of a quantifier (∀ or ∃).

The above example is a special case of a metatheorem called Ƚos’s principle.

Theorem 1.5 (Ƚos’s principle) Let (R_α)_α∈I be a family of rings, (resp. fields, modules, ... ) and ℱ an ultrafilter on I. A first order sentence σ in the language of rings (resp. fields, modules, ... )...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction. Ultraproducts. Definitions and examples
• 2 Elementary equivalence. Axiomatizable and finitely axiomatizable classes. Examples and results in field theory
• 3 Elementary definability. Applications to polynomial and power series rings and their quotient fields
• 4 Peano rings and Peano fields
• 5 Hilbertian fields and realizations of finite groups as Galois groups
• 6 The language of modules over a fixed ring
• 7 Algebraically compact modules
• 8 Decompositions and algebraic compactness
• 9 The two-sorted language of modules over unspecified rings
• 10 The first order theory of rings
• 11 Pure global dimension and algebraically compact rings
• 12 Representation theory of finite dimensional algebras
• 13 Problems
• Tables
• A Basic notions and definitions from homological algebra
• B Functor categories on finitely presented modules
• C Glossary of some basic notions in ring and module theory
• Bibliography
• Author Index
• Subject Index