Ring Theory

This textbook is designed for the UG/PG students of mathematics for all universities over the world. It is primarily based on the classroom lectures, the authors gave at the University of Delhi. This book is used both for self-study and course text. Full details of all proofs are included along with innumerous solved problems, interspersed throughout the text and at places where they naturally arise, to understand abstract notions. The proofs are precise and complete, backed up by chapter end problems, with just the right level of difficulty, without compromising the rigor of the subject. The book starts with definition and examples of Rings and logically follows to cover Properties of Rings, Subrings, Fields, Characteristic of a Ring, Ideals, Integral Domains, Factor Rings, Prime Ideals, Maximal Ideals and Primary Ideals, Ring Homomorphisms and Isomorphisms, Polynomial Rings, Factorization of Polynomials, and Divisibility in Integral Domains. Preface Foreword Contents List of Symbols 1 Rings 1.1 Definition and Examples of Rings 1.2 Elementary Properties of Rings 1.3 S Subrings 1.4 A Algebra of Subrings 1.5 Idempotent and Nilpotent Elements 2 Integral Domains and Fields 2.1 Special Kinds of Rings 2.2 Some Theorems on Integral Domains and Fields 2.3 Characteristic of a Ring 3 Ideals and Factor Rings 3.1 Ideals in a Ring 3.2 Intersection and Union of Ideals 3.3 Sum and Product of Two Ideals 3.4 Ideal Generated by a Subset 3.5 Simple Rings 3.6 Factor Rings 3.7 Types of Ideals 4 Ring Homomorphisms and Isomorphisms 4.1 Ring Homomorphism 4.2 Properties of Ring Homomorphisms 4.3 Kernel of Ring Homomorphism 4.4 Applications of Natural Homomorphism 4.5 Isomorphism Theorems 4.6 The Field of Quotients of an Integral Domain 5 Polynomial Rings 5.1 Ring of Polynomials 5.2 The Division Algorithm and its Consequences 5.3 Principal Ideal Domain 6 Factorization of Polynomials 6.1 Irreducible and Reducible Polynomials 6.2 Irreducibility Tests 6.3 Irreducible Polynomials, Maximal Ideals and Fields 7 Divisibility in Integral Domains 7.1 Irreducible and Prime Elements 7.2 Unique Factorization Domains 7.3 Euclidean Domains Appendix One Index