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

Subject Matter

 * Ring Theory
 * Module Theory
 * Linear Algebra

Part I: Rings and Modules

 * Rings - definitions and examples
 * The definition of a ring
 * Some examples of rings
 * Some special classes of rings


 * Subrings, homomorphisms and ideals
 * Subrings
 * Homomorphisms
 * Some properties of subrings and ideals


 * Construction of new rings
 * Direct sums
 * Polynomial rings
 * Matrix rings


 * Factorization in integral domains
 * Integral domains
 * Divisors, units and associates
 * Unique factorization domains
 * Principal ideal domains and Euclidean domains
 * More about Euclidean domains


 * Modules
 * The definition of a module over a ring
 * Submodules
 * Homomorphisms and quotient modules
 * Direct sums of modules


 * Some special classes of modules
 * More on finitely-generated modules
 * Torsion modules
 * Free modules

Part II: Direct Decompositon of a Finitely-Generated Module over a Principal Ideal Domain

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


 * Decomposition theorems
 * The main theorem
 * Uniqueness of the decomposition
 * The primary decomposition of a module


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

Part III: Applications to Groups and Matrices

 * Finitely-generated Abelian groups
 * $$\Z$$-modules
 * Classification of finitely-generated Abelian groups
 * Finite Abelian groups
 * Generators and relations
 * Computing invariants from presentations


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


 * Computation of canonical forms
 * The module formulation
 * The kernel of $$\epsilon$$
 * The rational canonical form
 * The primary rational and Jordan canonical forms

Cited by

 * $$\S 62.2$$