Textbook in PDF format

This textbook is designed for a first course in ring theory, module theory and category theory. Written following several decades of teaching experience, it stands out with its clear and engaging style, featuring thorough explanations and attention to detail. Carefully selected exercises encourage active learning and problem-solving.
The textbook integrates elementary category theory with basic concepts and examples developed throughout the course. Although the primary focus is on rings and modules, relevant notions for other algebraic structures, such as groups and semigroups, are also discussed. Thus, this book aims at introducing students to noncommutative rings and modules within a broader algebraic context.
Aimed at advanced undergraduates or master students in mathematics, this textbook is suitable both for use in the classroom and self-study. Whereas the first part of the book covers a basic course in ring and module theory, the latter part includes optional deepening topics.
Preface
Contents
Basic Notions
Rings
Categories
Functors
Modules
Bimodules
Submodules and Quotients
Natural Transformations of Functors
Sums and Intersections, Direct Sums and Direct Products
Homomorphisms Between Finite Direct Sums of Modules
Isomorphism Theorems
Sets of Generators, Maximal Submodules
The Lattice of Submodules
Some Classes of Modules
Free Modules
Further Properties of Free Modules
Other Free Algebraic Structures
Free Monoids
Free Groups
Free k-Algebras
Further Terminology About Categories
Every Category Has a Skeleton
IBN Rings
Exact Sequences
Projective Modules
Tensor Product of Modules
Tensor Product of Morphisms
Projective Z-Modules: Hereditary Rings
Idempotents, Nilpotents, Further Results on Projective Modules
Simple Modules, Semisimple Modules
Noetherian Modules, Artinian Modules
Modules Over the Factor Ring R/I
Series of Modules: Modules of Finite Composition Length
Right Artinian Rings
Semisimple Artinian Rings
The Ring of nn Matrices Over a Division Ring
Example of Ring That Is Left Hereditary But Not Right Hereditary
The Artin-Wedderburn Theorem
The Nilradical, Right Artinian Rings, and the Hopkins-Levitzki Theorem
The Radical of a Module
The Jacobson Radical of a Ring
Group Representations
Local Rings, Injective Modules, Flat Modules
Local Rings
Injective Modules
Projective Covers
Injective Envelopes
Uniform Modules, Goldie Dimension
Direct Limits of Modules
Direct Limit as a Functor
Direct Limits of Rings
Flat Modules
Finitely Presented Modules
Inverse Limits
Additive Categories, Abelian Categories
Monomorphisms, Epimorphisms, Subobjects, Quotient Objects
Products and Coproducts
Initial Objects, Preadditive Categories
Additive Categories
Equalizers, Coequalizers, Kernels and Cokernels
Abelian Categories
Exercises on Abelian Categories
Appendices
Appendix : Axiomatic Set Theory
Appendix : Cardinal Numbers, Ordinal Numbers
Cardinal numbers
Partial Order for Cardinal Numbers
Partially Ordered Sets
Ordinals
Initial Segments of Well-ordered Sets
Formal Definition of Ordinal Numbers
Successor Ordinals, Limit Ordinals
Arithmetic of Ordinal Numbers
Exponentiation
For Further Study…
Bibliography
Index