Model theory and modules /:
In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge, England ; New York :
Cambridge University Press,
1988.
|
| Series: | London Mathematical Society lecture note series ;
130. |
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time. |
| Item Description: | Includes indexes. |
| Physical Description: | 1 online resource (xviii, 380 pages) : illustrations |
| Bibliography: | Includes bibliographical references (pages 351-369). |
| ISBN: | 9781107361430 1107361435 9780511600562 0511600569 |
Staff View
MARC
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| 245 | 1 | 0 | |a Model theory and modules / |c Mike Prest. |
| 260 | |a Cambridge, England ; |a New York : |b Cambridge University Press, |c 1988. | ||
| 300 | |a 1 online resource (xviii, 380 pages) : |b illustrations | ||
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| 490 | 1 | |a London Mathematical Society lecture note series ; |v 130 | |
| 504 | |a Includes bibliographical references (pages 351-369). | ||
| 500 | |a Includes indexes. | ||
| 588 | 0 | |a Print version record. | |
| 520 | |a In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time. | ||
| 505 | 0 | |a Cover; Title; Copyright; Dedication; Preface; Contents; Introduction; Acknowledgements; Notations and conventions; Remarks on the development of the area; Section summaries; Chapter 1 Some preliminaries; 1.1 An introduction to model theory; 1.2 Injective modules and decomposition theorems; Chapter 2 Positive primitive formulas and the sets they define; 2.1 pp formulas; 2.2 pp-types; 2.3 Pure embeddings and pure-injective modules; 2.4 pp-elimination of quantifiers; 2.5 Immediate corollaries of pp-elimination of quantifiers; 2.6 Comparison of complete theories of modules | |
| 505 | 8 | |a 2.Z pp formulas and types in abelian groups2.L Other languages; Chapter 3 Stability and totally transcendental modules; 3.1 Stability for modules; 3.2 A structure theorem for totally transcendental modules; part I; 3.A Abelian structures; Chapter 4 Hulls; 4.1 pp-essential embeddings and the construction of hulls; 4.2 Examples of hulls; 4.3 Decomposition of injective and pure-injective modules; 4.4 Irreducible types; 4.5 Limited and unlimited types; 4.6 A structure theory for totally transcendental modules; part II; 4.C Categoricity; 4.7 The space of indecomposables | |
| 505 | 8 | |a Chapter 5 Forking and ranks5.1 Forking and independence; 5.2 Ranks; 5.3 An algebraic characterisation of independence; 5.4 Independence when T = TXo; Chapter 6 Stability-theoretic properties of types; 6.1 Free parts of types and the stratified order; 6.2 Domination and the RK-order; 6.3 Orthogonality and the RK-order; 6.4 Regular types; 6.1 An example: injective modules over noetherian rings; 6.5 Saturation and pure-injective modules; 6.6 Multiplicity and strong types; Chapter 7 Superstable modules; 7.1 Superstable modules: the uncountable spectrum; 7.2 Modules of U-rank 1 | |
| 505 | 8 | |a 7.3 Modules of finite U-rankChapter 8 The lattice of pp-types and free realisations of pp-types; 8.1 The lattice of pp-types; 8.2 Finitely generated pp-types; 8.3 pp-types and matrices; 8.4 Duality and pure-semisimple rings; Chapter 9 Types and the structure of pure-injective modules; 9.1 Minimal pairs; 9.2 Associated types; 9.3 Notions of isolation; 9.4 Neg-isolated types and elementary cogenerators; Chapter 10 Dimension and decomposition; 10.1 Existence of indecomposable direct summands; 10.2 Dimensions defined on lattices; 10.3 Modules with width | |
| 505 | 8 | |a 10.4 Classification for theories with dimension10.5 Krull dimension; 10.T Teq; 10.6 Dimension and height; 10.V Valuation rings; Chapter 11 Modules over artinian rings; 11.1 Pure-semisimple rings; 11.2 Pure-semisimple rings and rings of finite representation type; 11.3 Finite hulls over artinian rings; 11.4 Finite Morley rank and finite representation type; 11.P Pathologies -- Chapter 12 Functor categories; 12.1 Functors defined from pp formulas; 12.2 Simple functors; 12.3 Embedding into functor categories; 12.P Pure global dimension and dimensions of functor categories | |
| 650 | 0 | |a Model theory. |0 http://id.loc.gov/authorities/subjects/sh85086421 | |
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| 650 | 7 | |a Modules (algèbre) |2 ram | |
| 776 | 0 | 8 | |i Print version: |a Prest, Mike. |t Model theory and modules. |d Cambridge, England ; New York : Cambridge University Press, 1988 |z 0521348331 |w (DLC) 87032640 |w (OCoLC)17108169 |
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| contents | Cover; Title; Copyright; Dedication; Preface; Contents; Introduction; Acknowledgements; Notations and conventions; Remarks on the development of the area; Section summaries; Chapter 1 Some preliminaries; 1.1 An introduction to model theory; 1.2 Injective modules and decomposition theorems; Chapter 2 Positive primitive formulas and the sets they define; 2.1 pp formulas; 2.2 pp-types; 2.3 Pure embeddings and pure-injective modules; 2.4 pp-elimination of quantifiers; 2.5 Immediate corollaries of pp-elimination of quantifiers; 2.6 Comparison of complete theories of modules 2.Z pp formulas and types in abelian groups2.L Other languages; Chapter 3 Stability and totally transcendental modules; 3.1 Stability for modules; 3.2 A structure theorem for totally transcendental modules; part I; 3.A Abelian structures; Chapter 4 Hulls; 4.1 pp-essential embeddings and the construction of hulls; 4.2 Examples of hulls; 4.3 Decomposition of injective and pure-injective modules; 4.4 Irreducible types; 4.5 Limited and unlimited types; 4.6 A structure theory for totally transcendental modules; part II; 4.C Categoricity; 4.7 The space of indecomposables Chapter 5 Forking and ranks5.1 Forking and independence; 5.2 Ranks; 5.3 An algebraic characterisation of independence; 5.4 Independence when T = TXo; Chapter 6 Stability-theoretic properties of types; 6.1 Free parts of types and the stratified order; 6.2 Domination and the RK-order; 6.3 Orthogonality and the RK-order; 6.4 Regular types; 6.1 An example: injective modules over noetherian rings; 6.5 Saturation and pure-injective modules; 6.6 Multiplicity and strong types; Chapter 7 Superstable modules; 7.1 Superstable modules: the uncountable spectrum; 7.2 Modules of U-rank 1 7.3 Modules of finite U-rankChapter 8 The lattice of pp-types and free realisations of pp-types; 8.1 The lattice of pp-types; 8.2 Finitely generated pp-types; 8.3 pp-types and matrices; 8.4 Duality and pure-semisimple rings; Chapter 9 Types and the structure of pure-injective modules; 9.1 Minimal pairs; 9.2 Associated types; 9.3 Notions of isolation; 9.4 Neg-isolated types and elementary cogenerators; Chapter 10 Dimension and decomposition; 10.1 Existence of indecomposable direct summands; 10.2 Dimensions defined on lattices; 10.3 Modules with width 10.4 Classification for theories with dimension10.5 Krull dimension; 10.T Teq; 10.6 Dimension and height; 10.V Valuation rings; Chapter 11 Modules over artinian rings; 11.1 Pure-semisimple rings; 11.2 Pure-semisimple rings and rings of finite representation type; 11.3 Finite hulls over artinian rings; 11.4 Finite Morley rank and finite representation type; 11.P Pathologies -- Chapter 12 Functor categories; 12.1 Functors defined from pp formulas; 12.2 Simple functors; 12.3 Embedding into functor categories; 12.P Pure global dimension and dimensions of functor categories |
| ctrlnum | (OCoLC)839304954 |
| dewey-full | 511/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.8 |
| dewey-search | 511/.8 |
| dewey-sort | 3511 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn839304954 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:41:21Z |
| institution | BVB |
| isbn | 9781107361430 1107361435 9780511600562 0511600569 |
| language | English |
| oclc_num | 839304954 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (xviii, 380 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | London Mathematical Society lecture note series ; |
| series2 | London Mathematical Society lecture note series ; |
| spelling | Prest, Mike. Model theory and modules / Mike Prest. Cambridge, England ; New York : Cambridge University Press, 1988. 1 online resource (xviii, 380 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society lecture note series ; 130 Includes bibliographical references (pages 351-369). Includes indexes. Print version record. In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time. Cover; Title; Copyright; Dedication; Preface; Contents; Introduction; Acknowledgements; Notations and conventions; Remarks on the development of the area; Section summaries; Chapter 1 Some preliminaries; 1.1 An introduction to model theory; 1.2 Injective modules and decomposition theorems; Chapter 2 Positive primitive formulas and the sets they define; 2.1 pp formulas; 2.2 pp-types; 2.3 Pure embeddings and pure-injective modules; 2.4 pp-elimination of quantifiers; 2.5 Immediate corollaries of pp-elimination of quantifiers; 2.6 Comparison of complete theories of modules 2.Z pp formulas and types in abelian groups2.L Other languages; Chapter 3 Stability and totally transcendental modules; 3.1 Stability for modules; 3.2 A structure theorem for totally transcendental modules; part I; 3.A Abelian structures; Chapter 4 Hulls; 4.1 pp-essential embeddings and the construction of hulls; 4.2 Examples of hulls; 4.3 Decomposition of injective and pure-injective modules; 4.4 Irreducible types; 4.5 Limited and unlimited types; 4.6 A structure theory for totally transcendental modules; part II; 4.C Categoricity; 4.7 The space of indecomposables Chapter 5 Forking and ranks5.1 Forking and independence; 5.2 Ranks; 5.3 An algebraic characterisation of independence; 5.4 Independence when T = TXo; Chapter 6 Stability-theoretic properties of types; 6.1 Free parts of types and the stratified order; 6.2 Domination and the RK-order; 6.3 Orthogonality and the RK-order; 6.4 Regular types; 6.1 An example: injective modules over noetherian rings; 6.5 Saturation and pure-injective modules; 6.6 Multiplicity and strong types; Chapter 7 Superstable modules; 7.1 Superstable modules: the uncountable spectrum; 7.2 Modules of U-rank 1 7.3 Modules of finite U-rankChapter 8 The lattice of pp-types and free realisations of pp-types; 8.1 The lattice of pp-types; 8.2 Finitely generated pp-types; 8.3 pp-types and matrices; 8.4 Duality and pure-semisimple rings; Chapter 9 Types and the structure of pure-injective modules; 9.1 Minimal pairs; 9.2 Associated types; 9.3 Notions of isolation; 9.4 Neg-isolated types and elementary cogenerators; Chapter 10 Dimension and decomposition; 10.1 Existence of indecomposable direct summands; 10.2 Dimensions defined on lattices; 10.3 Modules with width 10.4 Classification for theories with dimension10.5 Krull dimension; 10.T Teq; 10.6 Dimension and height; 10.V Valuation rings; Chapter 11 Modules over artinian rings; 11.1 Pure-semisimple rings; 11.2 Pure-semisimple rings and rings of finite representation type; 11.3 Finite hulls over artinian rings; 11.4 Finite Morley rank and finite representation type; 11.P Pathologies -- Chapter 12 Functor categories; 12.1 Functors defined from pp formulas; 12.2 Simple functors; 12.3 Embedding into functor categories; 12.P Pure global dimension and dimensions of functor categories Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Modules (Algebra) http://id.loc.gov/authorities/subjects/sh85086470 Théorie des modèles. Modules (Algèbre) MATHEMATICS General. bisacsh Model theory fast Modules (Algebra) fast Modèles, Théorie des. ram Modules (algèbre) ram Print version: Prest, Mike. Model theory and modules. Cambridge, England ; New York : Cambridge University Press, 1988 0521348331 (DLC) 87032640 (OCoLC)17108169 London Mathematical Society lecture note series ; 130. http://id.loc.gov/authorities/names/n42015587 |
| spellingShingle | Prest, Mike Model theory and modules / London Mathematical Society lecture note series ; Cover; Title; Copyright; Dedication; Preface; Contents; Introduction; Acknowledgements; Notations and conventions; Remarks on the development of the area; Section summaries; Chapter 1 Some preliminaries; 1.1 An introduction to model theory; 1.2 Injective modules and decomposition theorems; Chapter 2 Positive primitive formulas and the sets they define; 2.1 pp formulas; 2.2 pp-types; 2.3 Pure embeddings and pure-injective modules; 2.4 pp-elimination of quantifiers; 2.5 Immediate corollaries of pp-elimination of quantifiers; 2.6 Comparison of complete theories of modules 2.Z pp formulas and types in abelian groups2.L Other languages; Chapter 3 Stability and totally transcendental modules; 3.1 Stability for modules; 3.2 A structure theorem for totally transcendental modules; part I; 3.A Abelian structures; Chapter 4 Hulls; 4.1 pp-essential embeddings and the construction of hulls; 4.2 Examples of hulls; 4.3 Decomposition of injective and pure-injective modules; 4.4 Irreducible types; 4.5 Limited and unlimited types; 4.6 A structure theory for totally transcendental modules; part II; 4.C Categoricity; 4.7 The space of indecomposables Chapter 5 Forking and ranks5.1 Forking and independence; 5.2 Ranks; 5.3 An algebraic characterisation of independence; 5.4 Independence when T = TXo; Chapter 6 Stability-theoretic properties of types; 6.1 Free parts of types and the stratified order; 6.2 Domination and the RK-order; 6.3 Orthogonality and the RK-order; 6.4 Regular types; 6.1 An example: injective modules over noetherian rings; 6.5 Saturation and pure-injective modules; 6.6 Multiplicity and strong types; Chapter 7 Superstable modules; 7.1 Superstable modules: the uncountable spectrum; 7.2 Modules of U-rank 1 7.3 Modules of finite U-rankChapter 8 The lattice of pp-types and free realisations of pp-types; 8.1 The lattice of pp-types; 8.2 Finitely generated pp-types; 8.3 pp-types and matrices; 8.4 Duality and pure-semisimple rings; Chapter 9 Types and the structure of pure-injective modules; 9.1 Minimal pairs; 9.2 Associated types; 9.3 Notions of isolation; 9.4 Neg-isolated types and elementary cogenerators; Chapter 10 Dimension and decomposition; 10.1 Existence of indecomposable direct summands; 10.2 Dimensions defined on lattices; 10.3 Modules with width 10.4 Classification for theories with dimension10.5 Krull dimension; 10.T Teq; 10.6 Dimension and height; 10.V Valuation rings; Chapter 11 Modules over artinian rings; 11.1 Pure-semisimple rings; 11.2 Pure-semisimple rings and rings of finite representation type; 11.3 Finite hulls over artinian rings; 11.4 Finite Morley rank and finite representation type; 11.P Pathologies -- Chapter 12 Functor categories; 12.1 Functors defined from pp formulas; 12.2 Simple functors; 12.3 Embedding into functor categories; 12.P Pure global dimension and dimensions of functor categories Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Modules (Algebra) http://id.loc.gov/authorities/subjects/sh85086470 Théorie des modèles. Modules (Algèbre) MATHEMATICS General. bisacsh Model theory fast Modules (Algebra) fast Modèles, Théorie des. ram Modules (algèbre) ram |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85086421 http://id.loc.gov/authorities/subjects/sh85086470 |
| title | Model theory and modules / |
| title_auth | Model theory and modules / |
| title_exact_search | Model theory and modules / |
| title_full | Model theory and modules / Mike Prest. |
| title_fullStr | Model theory and modules / Mike Prest. |
| title_full_unstemmed | Model theory and modules / Mike Prest. |
| title_short | Model theory and modules / |
| title_sort | model theory and modules |
| topic | Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Modules (Algebra) http://id.loc.gov/authorities/subjects/sh85086470 Théorie des modèles. Modules (Algèbre) MATHEMATICS General. bisacsh Model theory fast Modules (Algebra) fast Modèles, Théorie des. ram Modules (algèbre) ram |
| topic_facet | Model theory. Modules (Algebra) Théorie des modèles. Modules (Algèbre) MATHEMATICS General. Model theory Modèles, Théorie des. Modules (algèbre) |
| work_keys_str_mv | AT prestmike modeltheoryandmodules |