This book, "Foundations of Module and Ring Theory: A Handbook for Study and Research," is the third volume in the "Algebra, Logic, and Applications" series, edited by R. Göbel and A. Macintyre. It is authored by Robert Wisbauer and is a revised and updated English edition of the German book "Grundlagen der Modul- und Ringtheorie" published in 1988. The book provides a comprehensive introduction to module theory and related aspects of ring theory, starting from basic linear algebra. It includes recent results and new developments, making it a valuable resource for both researchers and students. The approach is categorical, and the book covers a wide range of topics, including: 1. ** Elementary Properties of Rings**: Definitions, properties of elements, ideals, and rings, including direct sums, ideals, and factor rings. 2. **Module Categories**: Elementary properties of modules, the category of \(R\)-modules, internal direct sums, products, coproducts, pullbacks, pushouts, functors, and tensor products. 3. **Modules Characterized by the Hom-Functor**: Generators, cogenerators, injective and projective modules, and essential extensions. 4. **Notions Derived from Simple Modules**: Semisimple modules and rings, socle and radical, and co-semisimple and good modules and rings. 5. **Finiteness Conditions in Modules**: Direct limits, finitely presented modules, coherent modules, Noetherian and Artinian modules, and annihilator conditions. 6. **Dual Finiteness Conditions**: Inverse limits, finitely copresented modules, Artinian and co-Noetherian modules, and modules of finite length. 7. **Pure Sequences and Derived Notions**: P-pure sequences and pure projective modules. 8. **Modules Described by means of Projectivity**: (Semi)hereditary modules and rings, semihereditary and hereditary domains, supplemented modules, semiperfect modules and rings, and perfect modules and rings. 9. **Relations between Functors**: Functorial morphisms, adjoint pairs of functors, equivalences of categories, dualities between categories, and quasi-Frobenius modules and rings. 10. **Functor Rings**: Rings with local units, global dimensions, functor rings of \(\sigma[M]\) and \(R\)-MOD, pure semisimple modules and rings, modules of finite representation type, serial modules and rings, and homo-serial modules and rings. The book is structured into chapters that cover various aspects of module and ring theory, providing detailed proofs and exercises to enhance understanding. It also includes a bibliography and an index for reference. The author, Robert Wisbauer, is from the University of Düsseldorf, Germany, and the book is published by Gordon and Breach Science Publishers.This book, "Foundations of Module and Ring Theory: A Handbook for Study and Research," is the third volume in the "Algebra, Logic, and Applications" series, edited by R. Göbel and A. Macintyre. It is authored by Robert Wisbauer and is a revised and updated English edition of the German book "Grundlagen der Modul- und Ringtheorie" published in 1988. The book provides a comprehensive introduction to module theory and related aspects of ring theory, starting from basic linear algebra. It includes recent results and new developments, making it a valuable resource for both researchers and students. The approach is categorical, and the book covers a wide range of topics, including: 1. ** Elementary Properties of Rings**: Definitions, properties of elements, ideals, and rings, including direct sums, ideals, and factor rings. 2. **Module Categories**: Elementary properties of modules, the category of \(R\)-modules, internal direct sums, products, coproducts, pullbacks, pushouts, functors, and tensor products. 3. **Modules Characterized by the Hom-Functor**: Generators, cogenerators, injective and projective modules, and essential extensions. 4. **Notions Derived from Simple Modules**: Semisimple modules and rings, socle and radical, and co-semisimple and good modules and rings. 5. **Finiteness Conditions in Modules**: Direct limits, finitely presented modules, coherent modules, Noetherian and Artinian modules, and annihilator conditions. 6. **Dual Finiteness Conditions**: Inverse limits, finitely copresented modules, Artinian and co-Noetherian modules, and modules of finite length. 7. **Pure Sequences and Derived Notions**: P-pure sequences and pure projective modules. 8. **Modules Described by means of Projectivity**: (Semi)hereditary modules and rings, semihereditary and hereditary domains, supplemented modules, semiperfect modules and rings, and perfect modules and rings. 9. **Relations between Functors**: Functorial morphisms, adjoint pairs of functors, equivalences of categories, dualities between categories, and quasi-Frobenius modules and rings. 10. **Functor Rings**: Rings with local units, global dimensions, functor rings of \(\sigma[M]\) and \(R\)-MOD, pure semisimple modules and rings, modules of finite representation type, serial modules and rings, and homo-serial modules and rings. The book is structured into chapters that cover various aspects of module and ring theory, providing detailed proofs and exercises to enhance understanding. It also includes a bibliography and an index for reference. The author, Robert Wisbauer, is from the University of Düsseldorf, Germany, and the book is published by Gordon and Breach Science Publishers.