Book:B. Hartley/Rings, Modules and Linear Algebra

Subject Matter

 * Ring Theory
 * Module Theory
 * Linear Algebra

Contents

 * Preface
 * Organization of Topics


 * Part $\text {I}$: Rings and Modules


 * 1. Rings - definitions and examples
 * 1. The definition of a ring
 * 2. Some examples of rings
 * 3. Some special classes of rings


 * 2. Subrings, homomorphisms and ideals
 * 1. Subrings
 * 2. Homomorphisms
 * 3. Some properties of subrings and ideals


 * 3. Construction of new rings
 * 1. Direct sums
 * 2. Polynomial rings
 * 3. Matrix rings


 * 4. Factorization in integral domains
 * 1. Integral domains
 * 2. Divisors, units and associates
 * 3. Unique factorization domains
 * 4. Principal ideal domains and Euclidean domains
 * 5. More about Euclidean domains


 * 5. Modules
 * 1. The definition of a module over a ring
 * 2. Submodules
 * 3. Homomorphisms and quotient modules
 * 4. Direct sums of modules


 * 6. Some special classes of modules
 * 1. More on finitely-generated modules
 * 2. Torsion modules
 * 3. Free modules


 * Part $\text {II}$: Direct Decompositon of a Finitely-Generated Module over a Principal Ideal Domain


 * 7. Submodules of free modules
 * 1. The programme
 * 2. Free modules - bases, endomorphisms and matrices
 * 3. A matrix formulation of Theorem 7.1
 * 4. Elementary row and column operations
 * 5. Proof of 7.10 for Euclidean domains
 * 6. The general case
 * 7. Invariant factors
 * 8. Summary and a worked example


 * 8. Decomposition theorems
 * 1. The main theorem
 * 2. Uniqueness of the decomposition
 * 3. The primary decomposition of a module


 * 9. Decomposition theorems - a matrix-free approach
 * 1. Existence of the decompositions
 * 2. Uniqueness - a cancellation property of FG modules


 * Part $\text {III}$: Applications to Groups and Matrices


 * 10. Finitely-generated Abelian groups
 * 1. $\Z$-modules
 * 2. Classification of finitely-generated Abelian groups
 * 3. Finite Abelian groups
 * 4. Generators and relations
 * 5. Computing invariants from presentations


 * 11. Linear transformations, matrices and canonical forms
 * 1. Matrices and linear transformations
 * 2. Invariant subspaces
 * 3. $V$ as a $\mathbf k \left[{x}\right]$ module
 * 4. Matrices for cyclic linear transformations
 * 5. Canonical forms
 * 6. Minimal and characteristic polynomials


 * 12. Computation of canonical forms
 * 1. The module formulation
 * 2. The kernel of $\epsilon$
 * 3. The rational canonical form
 * 4. The primary rational and Jordan canonical forms


 * References


 * Index



Source work progress
* : $\S 3.2$: Polynomial rings: Lemma $3.8$


 * Redoing from start: examples from here to be done, also revisit Ring Examples to determine exactly what ring types they are.


 * : Chapter $1$: Rings - Definitions and Examples: Exercise $1$