Ring theory is a crucial area of abstract algebra that is essential for understanding advanced mathematics. In the book “Ring Theory” by Louis Halle Rowen, students will find a comprehensive guide to the fundamental concepts and principles of ring theory. This book is designed to help students understand the underlying structure of rings, and to provide them with the knowledge and skills they need to succeed in advanced mathematics.

One of the key features of the Ring Theory book is its clear and concise writing style. The author has done an excellent job of breaking down complex mathematical concepts into easy-to-understand terms and explanations, making the book ideal for students of all levels. Additionally, the book is filled with a variety of examples and practice problems that help students reinforce what they have learned and gain a deeper understanding of the material.

Another important aspect of the Ring Theory book is its comprehensive approach to teaching ring theory. The book covers all of the fundamental concepts that students will need to know in order to succeed in advanced mathematics and beyond. Whether a student is just starting to learn ring theory or has been studying it for a while, the Ring Theory book has something to offer.

**TABLE OF CONTENTS**

**Chapter 1 Construction of Rings**

**1.1 Matrix Rings and Idempotents**

- Matrices and Matrix Units
- Subrings of Matrix Rings
- Matrices Whose Entries Are Not Necessarily in Rings
- Idempotents and the Peirce Decomposition
- Idempotents and Simple Modules
- Primitive Idempotents
- Lifting Matrix Units and Idempotents

**1.2 Polynomial Rings**

- Monoid Rings
- Polynomial Rings
- Rings of Formal Power Series and of Laurent Series

**1.3 Free Modules and Rings**

- Free Modules
- Independent Modules and Direct Sums
- Modules Over Division Rings Are Free
- Free Objects
- Free Rings and Algebras
- Free Commutative Ring
- Diagrams
- Universals
- Invariant Base Number
- Weakly Finite Rings

**1.4 Products and Sums**

- Direct Products and Direct Sums
- Exact Sequences
- Split Exact Sequences
- Reduced Products
- Ultraproducts
- Products and Coproducts

**1.5 Endomorphism Rings and the Regular Representation**

- Endomorphism Rings
- Endomorphisms as Matrices
- The Dual Base
- Adjunction of 1

.

**1.6 Automorphisms, Derivations, and Skew Polynomial Rings**

- Automorphisms
- Derivations, Commutators, and Lie Algebras
- Skew Polynomial Rings and Ore Extensions
- Principal Left Ideal Domains (PLID’s)
- Skew Polynomial Rings (Without Derivation) Over Fields
- Differential Polynomial Rings Over Fields
- The Weyl Algebra
- Skew Power Series and Skew Laurent Series
- Skew Group Rings

**1.7 Tensor Products**

- Tensor Products of Bimodules and of Algebras
- Properties of the Tensor Operation
- Tensors and Centralizing Extensions
- Tensor Products Over Fields
- Tensor Products and Bimodules

**1.8 Direct Limits and Inverse Limits**

- Direct Limits
- D Supplement: Projective Limits
- The Completion

**1.9 Graded Rings and Modules**

- Tensor Rings

**1.10 Central Localization (also, cf. §2.12.9ff)**

- Structure Passing From R to S-‘ R
- Examples of Central Localization
- Central Localization of Modules

**Chapter 2 Basic Structure Theory**

**2.1 Primitive Rings**

- Jacobson’s Density Theorem
- Prime Rings and Minimal Left Ideals
- Finite-Ranked Transformations and the Socle
- Examples of Primitive Rings
- E Supplement: A Right Primitive Ring Which Is Not Primitive

**2.2 The Chinese Remainder Theorem and Subdirect Products**

- Subdirect Products
- Serniprime Rings

**2.3 Modules with Composition Series and Artinian Rings**

- The Jordan-Holder and Schreier Theorems
- Artinian Rings
- Split Semisimple Artinian Algebras
- Central Simple Algebras and Splitting
- Zariski Topology (for Finite Dimensional Algebras)

**2.4 Completely Reducible Modules and the Socle**

**2.5 The Jacobson Radical**

- Quasi-Invertibility
- Idempotents and the Jacobson Radical
- Weak Nullstellensatz and the Jacobson Radical
- The Structure Theoretical Approach to Rings
- “Nakayama’s Lemma”
- The Radical of a Module
- F Supplement: Finer Results Concerning the Jacobson Radical
- Wedderburn’s Principal Theorem
- F Supplement: Amitsur’s Theorem and Graded Rings
- F Supplement: The Jacobson Radical of a Rng
- Galois Theory of Rings

**2.6 Nilradicals**

- Nilpotent Ideals and Nilradicals
- The Nilradical of Noetherian Rings
- Bounded Index
- Derivations and Nilradicals
- Nil Subsets
- G Supplement: Koethe’s Conjecture

**2.7 Semiprimary Rings and Their Generalizations**

- Chain Conditions on Principal Left Ideals
- Passing from R to eRe and Back
- Semiperfect Rings
- Structure of Idempotents of Rings
- H Supplement: Perfect Rings

**2.8 Projective Modules (An Introduction)**

- Projective Modules
- Projective Versus Free
- Hereditary Rings
- Dual Basis Lemma
- Flat Modules
- Schanuel’s Lemma and Finitely Presented Modules
- H Supplement: Projective Covers

**2.9 Indecomposable Modules and LE-Modules**

- The Krull-Schmidt Decomposition
- Uniqueness of the Krull-Schmidt Decomposition
- Applications of the Krull-Schmidt Theorem to
- Semiperfect Rings
- Decompositions of Modules Over Noetherian Rings
- C Supplement: Representation Theory

**2.10 Injective Modules**

- Divisible Modules
- Essential Extensions and the Injective Hull
- Criteria for Injectivity and Projectivity
- J Supplement: Krull-Schmidt Theory for
- Injective Modules

**2.11 Exact Functors**

- Flat Modules and Injectives
- Regular Rings

**2.12 The Prime Spectrum**

- Localization and the Prime Spectrum
- Localizing at Central Prime Ideals
- The Rank of a Projective Module
- K Supplement: Hilbert’s Nullstellensatz and Generic Flatness
- Comparing Prime Ideals of Related Rings
- LO and the Prime Radical
- Height of Prime Ideals

**Chapter 3 Rings of Fractions and Embedding Theorems**

**3.1 Classical Rings of Fractions**

- Properties of Fractions
- Localizations of Modules

**3.2 Goldie’s Theorems and Orders in Artinian Quotient Rings**

- Annihilators
- ACC (Ann)
- Goldie rings
- Goldie Rank and Uniform Dimension
- Annihilator Ideals in Semiprime Rings
- Digression: Orders in Semilocal Rings
- Embedding of Rings
- Universal Sentences and Embeddings
- Embeddings into Primitive and Semiprimitive Rings
- L Supplement: Embedding into Matrix Rings
- L Supplement: Embedding into Division Rings
- L Supplement: Embedding into Artinian Rings

**3.3 Localization of Nonsingular Rings and Their Modules**

- Johnson’s Ring of Quotients of a Nonsingular Ring
- The Injective Hull of a Nonsingular Module
- Carrying Structure of R to R

**3.4 Noncommutative Localization**

- Digression: The Maximal Ring of Quotients
- The Martindale-Amitsur Ring of Quotients
- C Supplement: Idempotent Filters and Serre Categories

**3.5 Left Noetherian Rings**

- Constructing Left Noetherian Rings
- The Reduced Rank
- The Principal Ideal Theorem
- Height of Prime Ideals
- Artinian Properties of Noetherian Rings: Prelude to Jacobson’s Conjecture
- D Supplement: Noetherian Completion of a Ring
- Krull Dimension for Noncommutative Left
- Noetherian Rings
- Other Krull Dimensions
- Critical Submodules
- N Supplement: Gabriel Dimension

**Chapter 4 Categorical Aspects of Module Theory**

**4.1 The Morita Theorems**

- Categorical Notions
- Digression: Two Examples
- Morita Contexts and Morita’s Theorems
- Proof of Morita’s Theorem and Applications Exercises

**Chapter 5 Homology and Cohomology**

**5.1 Resolutions and Projective and Injective Dimension**

- Projective Resolutions
- Elementary Properties of Projective Dimension
- The (Left) Global Dimension
- Stably Free and FFR
- 0 Supplement: Quillen’s Theorem
- Euler Characteristic

**5.2 Homology, Cohomology, and Derived Functors**

- Homology
- The Long Exact Sequence
- Cohomology
- Derived Functors
- Tor and Ext
- Digression: Ext and Module Extensions
- Homological Dimension and the Derived Functors
- Dimension Shifting
- Acyclic Complexes

**5.3 Separable Algebras and Azumaya Algebras**

- Separable Algebras
- Hochschild’s Cohomology
- Azumaya Algebras

**Chapter 6 Rings with Polynomial Identities and Affine Algebras**

**6.1 Rings with Polynomial Identities**

- Central Polynomials and Identities of Matrices
- Structure Theory for PI-Rings
- The Artin-Procesi Theorem
- The n-Spectrum
- The Algebra of Generic Matrices
- The Identities of a PI-Algebra
- Noetherian PI-Rings

**6.2 Affine Algebras**

- Kurosch’s Problem and the Golod-Shafarevich
- Counterexample
- Growth of Algebras
- Gelfand-Kirillov Dimension
- The GK Dimension of a Module

**6.3 Affine PI-Algebras**

- The Nullstellensatz
- Shirshov’s Theorem
- The Theory of Prime PI-Rings
- Nilpotence of the Jacobson Radical
- Dimension Theory of Affine PI-Algebras

**Chapter 7 Central Simple Algebras**

**7.1 Structure of Central Simple Algebras**

- Centralizers and Splitting Fields
- Examples of Central Simple Algebras
- Overview of the Theory of Central Simple Algebras
- The First Generic Constructions
- Algebras of Degree 2, 3, 4

**7.2 The Brauer Group**

- General Properties of the Brauer Group
- First Invariant of Br(F): The Index
- Second Invariant of Br(F): The Exponent

**Chapter 8 Rings from Representation Theory**

**8.1 General Structure Theory of Group Algebras**

- An Introduction to the Zero-Divisor Question
- Maschke’s Theorem and the Regular Representation
- Group Algebras of Subgroups
- Augmentation Ideals and the Characteristic p Case
- The Kaplansky Trace Map and Nil Left Ideals
- Subgroups of Finite Index
- Prime Group Rings
- F Supplement: The Jacobson Radical of Group Algebras

**8.2 Noetherian Group Rings**

- Polycyclic-by-Finite Groups
- Digression: Growth of Groups
- Homological Dimension of Group Algebras
- Traces and Projectives
- 8.3 Enveloping Algebras
- Basic Properties of Enveloping Algebras
- Nilpotent and Solvable Lie Algebras
- Solvable Lie Algebras as Iterated Differential Polynomial Rings
- Completely Prime Ideals (Following BGR)
- Dimension Theory on U(L)

**8.4 General Ring Theoretic Methods**

- Vector Generic Flatness and the Nullstellensatz
- The Primitive Spectrum
- Reduction Techniques

**Download Now Book in PDF format**

The Ring Theory book is also very user-friendly. It has been designed with students in mind and features a well-organized layout that makes it easy to find the information that you need. The book is also filled with helpful tips and tricks that make solving mathematical problems easier and more intuitive.

For students who are looking for additional resources to help them learn and understand ring theory, the Ring Theory book is available in PDF format for free download. This means that students can access the book from anywhere, at any time, and on any device. Whether they are studying at home, on the go, or in the classroom, they can have the Ring Theory book right at their fingertips.

In conclusion, Ring Theory by Louis Halle Rowen is an excellent resource for students who are looking to master the fundamental concepts of ring theory. With its clear and concise writing style, comprehensive approach to teaching, and user-friendly design, the Ring Theory book is the perfect tool for students who want to excel in mathematics. So, if you are looking for a reliable and effective guide to help you learn ring theory, be sure to check out the Ring Theory book today!