Invariance of Modules under Automorphisms of their Envelopes and Covers (London Mathematical Society Lecture Note Series, Series Number 466)

The theory of invariance of modules under automorphisms of their envelopes and covers has opened up a whole new direction in the study of module theory. It offers a new perspective on generalizations of injective, pure-injective and flat-cotorsion modules beyond relaxing conditions on liftings of homomorphisms. This has set off a flurry of work in the area, with hundreds of papers using the theory appearing in the last decade. This book gives the first unified treatment of the topic. The authors are real experts in the area, having played a major part in the breakthrough of this new theory and its subsequent applications. The first chapter introduces the basics of ring and module theory needed for the following sections, making it self-contained and suitable for graduate students. The authors go on to develop and explain their tools, enabling researchers to employ them, extend and simplify known results in the literature and to solve longstanding problems in module theory, many of which are discussed at the end of the book. "The study of modules which are invariant under the action of certain subsets of the endomorphism ring of their injective envelope can be drawn back to the pioneering work of Johnson and Wong in which they characterized quasi-injective modules as those modules which are invariant under any endomorphism of their injective envelope. Later, Dickson and Fuller studied modules which are invariant under the group of all automorphisms of their injective envelope and proved that any indecomposable automorphism-invariant module over an F-algebra A is quasi-injective provided that F is a field with more than two elements. But after that this topic remained in dormant stage for some time until Lee and Zhou picked it up again in their paper where they called such modules auto-invariant modules. But the major breakthrough on this topic came from two papers that appeared a few months later: one of them was a paper of Er, Singh and Srivastava where they proved that the automorphism-invariant modules are precisely the pseudo-injective modules studied earlier by Teply, Jain, Clark, Huynh and others. The other one was a paper by Guil Asensio, and Srivastava where they proved that automorphism-invariant modules satisfy the exchange property and also they provide a new class of clean modules. Soon after this Guil Asensio and Srivastava extended the result of Dickson and Fuller by proving that if A is an algebra over a field F with more than two elements, then a module over A is automorphism-invariant if and only if it is quasi-injective. In 2015, in a paper published in the Israel Journal of Mathematics, Guil Asensio, Tutuncu and Srivastava laid down the foundation of general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers. In this general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers, we have obtained many interesting properties of such modules and found examples of some important classes of modules. When this theory is applied to some particular situations, then we obtain results that extend and simplify several results existing in the literature. For example, as a consequence of these general results, one obtains that modules invariant under automorphisms of their injective (resp., pure-injective) envelopes satisfy the full exchange property. These results extend well-known results of Warfield, Fuchs, Huisgen-Zimmermann and Zimmermann. Most importantly, this study yields us a new tool and new perspective to look at generalizations of injective, pure-injective or at-cotorsion modules. Until now most of the generalizations of injective modules were focussed on relaxing conditions on lifting of homomorphisms but this theory has opened up a whole new direction in the study of module theory" -- Provided by publisher "The study of modules which are invariant under the action of certain subsets of the endomorphism ring of their injective envelope can be drawn back to the pioneering work of Johnson and Wong in which they characterized quasi-injective modules as those modules which are invariant under any endomorphism of their injective envelope. Later, Dickson and Fuller studied modules which are invariant under the group of all automorphisms of their injective envelope and proved that any indecomposable automorphism-invariant module over an F-algebra A is quasi-injective provided that F is a field with more than two elements. But after that this topic remained in dormant stage for some time until Lee and Zhou picked it up again in their paper where they called such modules auto-invariant modules. But the major breakthrough on this topic came from two papers that appeared a few months later: one of them was a paper of Er, Singh and Srivastava where they proved that the automorphism-invariant modules are precisely the pseudo-injective modules studied earlier by Teply, Jain, Clark, Huynh and others. The other one was a paper by Guil Asensio, and Srivastava where they proved that automorphism-invariant modules satisfy the exchange property and also they provide a new class of clean modules. Soon after this Guil Asensio and Srivastava extended the result of Dickson and Fuller by proving that if A is an algebra over a field F with more than two elements, then a module over A is automorphism-invariant if and only if it is quasi-injective. In 2015, in a paper published in the Israel Journal of Mathematics, Guil Asensio, Tutuncu and Srivastava laid down the foundation of general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers. In this general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers, we have obtained many interesting properties of such modules and found examples of some important classes of modules. When this theory is applied to some particular situations, then we obtain results that extend and simplify several results existing in the literature. For example, as a consequence of these general results, one obtains that modules invariant under automorphisms of their injective (resp., pure-injective) envelopes satisfy the full exchange property. These results extend well-known results of Warfield, Fuchs, Huisgen-Zimmermann and Zimmermann. Most importantly, this study yields us a new tool and new perspective to look at generalizations of injective, pure-injective or at-cotorsion modules. Until now most of the generalizations of injective modules were focussed on relaxing conditions on lifting of homomorphisms but this theory has opened up a whole new direction in the study of module theory" -- Résumé de l'éditeur Front matter Copyright Contents Preface 1 Preliminaries 1.1 Basics of Ring Theory and Module Theory 1.2 Simple and Semisimple Modules 1.3 Essential and Closed Submodules 1.4 Prime Rings and Semiprime Rings 1.5 Classical Rings of Fractions and Semiprime Goldie Rings 1.6 Local, Semilocal and Semiperfect Rings 1.7 Injective and Projective Modules 1.8 Injective Envelope and Quasi-Injective Modules 1.9 Flat Modules 1.10 Exchange Property of Modules 1.11 Pure-Injective and Cotorsion Modules 2 Modules Invariant under Antomorphisms of Envelopes 2.1 Introduction to Envelopes 2.2 Modules Invariant under Endomorphisms and Automorphisms 2.3 Additive Unit Structure of von Neumann Regular Rings 2.4 Applications of Additive Unit Structure of von Neumann Regular Rings 3 Structure and Properties of Modules Invariant under Automorphisms 3.1 Structure of Modules Invariant under Automorphisms 3.2 Properties of Modules Invariant under Automorphisms 3.3 Applications 3.4 Modules Invariant under Monomorphisms 4 Automorphism-Invariant Modules 4.1 Some Characterizations of Automorphism-Invariant Modules 4.2 Nonsingular Automorphism-lnvariant Rings 4.3 When Is an Automorphism-Invariant Module a Quasi-Injective Module 4.4 Rings Whose Cyclic Modules Are Automorphism-Invariant 4.5 Rings Whose Each One-Sided Ideal Is Automorphism-Invariant 5 Modules Coinvariant under Automorphisms of their Covers 5.1 Structure and Properties 5.2 Automorphism-Coinvariant Modules 5.3 Dual Automorphism-Invariant Modules 6 Schroder-Bernstein Problem 6.1 Schroder-Bernstein Problem for X-Endomorphism Invariant Modules 6.2 Schroder-Bernstein Problem for Automorphism-Invariant Modules 7 Automorphism-Extendable Modules 7.1 General Properties of Automorphism-Extendable Modules 7.2 Semi-Artinian Automorphism-Extendable Modules 7.3 Automorphism-Extendable Modules that Are Not Singular 7.4 Modules over Strongly Prime Rings 7.5 Modules over Hereditary Prime Rings 7.6 General Properties of Endomorphism-Extendable Modules and Rings 7.7 Annihilators that Are Ideals 7.8 Completely Integrally Closed Subrings and Self-Injective Rings 7.9 Endomorphism-Liftable and π-Projective Modules 7.10 Rings Whose Cyclic Modules are Endomorphism-Extendable 8 Automorpllism-Liftable Modules 8.1 Automorphism-Liftable and Endomorphism-Liftable Modules 8.2 Non-Primitive Hereditary Noetherian Prime Rings 8.3 Non-Primitive Dedekind Prime Rings 8.4 Idempotent-Lifted Modules and π-Projective Modules 9 Open Problems References Index This is the first book on the topic of study of modules invariant under automorphisms of their envelopes and covers. Containing plentiful examples and open problems, it is a valuable resource for graduate students and researchers in algebra who wish to learn the state of the art in this area of module theory.