Foundations of Module and Ring Theory: A Handbook for Study and Research by Robert Wisbauer is a comprehensive guide to the theoretical foundations of module and ring theory. This book provides a thorough treatment of the subject matter, making it an ideal resource for students and researchers alike.
The book begins with an introduction to the basic concepts of modules and rings, including the definition of a module and the definition of a ring. It then delves into the more advanced topics of the theory, such as submodules, factor modules, and simple modules. The book also covers the theory of ideals, including prime and maximal ideals, and the theory of quotient modules.
One of the strengths of this book is its use of numerous examples and exercises throughout the text. These examples help to illustrate the key concepts and bring the theory to life for the reader. The exercises also provide valuable opportunities for students to practice what they have learned and to test their understanding of the material.
TABLE OF CONTENTS
Chapter 1 Elementary properties of rings
- Basic notions
- Special elements and ideals in rings
- Special rings
- Chain conditions for rings
- Algebras and group rings
Chapter 2 Module categories
- Elementary properties of modules
- The category of R-modules
- Internal direct sum
- Product, coproduct and subdirect product
- Pullback and pushout
- Functors, Hom-functors
- Tensor product, tensor
Chapter 3 Modules characterized by the Hom-functor
- Generators, trace
- Cogenerators, reject
- Subgenerators, the category σ[M]
- Injective modules
- Essential extensions, injective hulls
- Projective modules
- Superfluous epimorphisms, projective covers
.
Chapter 4 Notions derived from simple modules
- Semisimple modules and rings
- Socle and radical of modules and rings
- The radical of endomorphism rings
- Co-semisimple and good modules and rings
Chapter 5 Finiteness conditions in modules
- The direct limit
- Finitely presented modules
- Coherent modules and rings
- Noetherian modules and rings
- Annihilator conditions
Chapter 6 Dual finiteness conditions
- The inverse limit
- Finitely copresented modules
- Artinian and co-noetherian modules
- Modules of finite length
Chapter 7 Pure sequences and derived notions
- P-pure sequences, pure projective modules
- Purity in σ[M], R-MOD and ZZ-MOD
- Absolutely pure modules
- Flat modules
- Regular modules and rings
- Copure sequences and derived notions
Chapter 8 Modules described by means of projectivity
- (Semi)hereditary modules and rings
- Semihereditary and hereditary domains
- Supplemented modules
- Semiperfect modules and rings
- Perfect modules and rings
Chapter 9 Relations between functors
- Functorial morphisms
- Adjoint pairs of functors
- Equivalences of categories
- Dualities between categories
- Quasi-Frobenius modules and rings
Chapter 10 Functor rings
- Rings with local units
- Global dimensions of modules and rings
- The functor Homb (V, −)
- Functor rings of σ[M] and R-MOD
- Pure semisimple modules and rings
- Modules of finite representation type
- Serial modules and rings
- Homo-serial modules and rings
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Another standout feature of this book is its treatment of homological algebra. The book provides a detailed overview of the basics of homological algebra, including the definition of chain complexes, homology and cohomology groups, and derived functors. This makes the book an excellent resource for those who want to deepen their understanding of this important area of mathematics.
In addition to its comprehensive treatment of the subject matter, Foundations of Module and Ring Theory also includes a number of appendices that provide additional information on related topics. These appendices include a discussion of torsion modules, flat modules, and injective and projective modules, among others.
In conclusion, Foundations of Module and Ring Theory by Robert Wisbauer is an essential resource for students and researchers studying module and ring theory. With its comprehensive treatment of the subject matter and its use of numerous examples and exercises, this book is sure to help students develop a deep understanding of the theory and its applications.
We are proud to announce that we have this book in PDF format available for download. So, students can easily access this valuable resource from the comfort of their own home.
