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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