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

## B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra

Published $1970$, Chapman and Hall

ISBN 0 412 09810 5.

### Contents

Preface
Organization of Topics
Part 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. 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 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 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