Rings, modules and linear algebra : a further course in algebra describing the structure of Abelian groups and canonical forms of matrices through the study of rings and modules

Title page......Page 1 Preface......Page 3 Organization of topics......Page 5 Contents......Page 7 PART I - RINGS AND MODULES......Page 11 1. The definition of a ring......Page 13 2. Some examples of rings......Page 15 3. Some special classes of rings......Page 21 1. Subrings......Page 25 2. Homomorphisms......Page 28 3. Some properties of subrings and ideals......Page 36 1. Direct sums......Page 43 2. Polynomial rings......Page 47 3. Matrix rings......Page 54 1. Integral domains......Page 59 2. Divisors, units and associates......Page 61 3. Unique factorization domains......Page 64 4. Principal ideal domains and Euclidean domains......Page 69 5. More about Euclidean domains......Page 72 1. The definition of a module over a ring......Page 79 2. Submodules......Page 84 3. Homomorphisms and quotient modules......Page 87 4. Direct sums of modules......Page 90 1. More on finitely-generated modules......Page 95 2. Torsion modules......Page 97 3. Free modules......Page 99 PART II - DIRECT DECOMPOSITION OF A FINITELY-GENERATED MODULE OVER A PRINCIPAL IDEAL DOMAIN......Page 107 1. The programme......Page 109 2. Free modules - bases, endomorphisms and matrices......Page 110 3. A matrix formulation of Theorem 7-1......Page 116 4. Elementary row and column operations......Page 120 5. Proof of 7.10 for Euclidean domains......Page 121 6. The general case......Page 124 7. Invariant factors......Page 125 8. Summary and a worked example......Page 128 1. The main theorem......Page 133 2. Uniqueness of the decomposition......Page 137 3. The primary decomposition of a module......Page 142 1. Existence of the decompositions......Page 149 2. Uniqueness - a cancellation property of FG modules......Page 153 PART III - APPLICATIONS TO GROUPS AND MATRICES......Page 159 1. Z-modules......Page 161 2. Classification of finitely-generated Abelian groups......Page 162 3. Finite Abelian groups......Page 164 4. Generators and relations......Page 167 5. Computing invariants from presentations......Page 170 1. Matrices and linear transformations......Page 177 2. Invariant subspaces......Page 179 3. V as a k[x]-module......Page 180 4. Matrices for cyclic linear transformations......Page 186 5. Canonical forms......Page 189 6. Minimal and characteristic polynomials......Page 194 1. The module formulation......Page 203 2. The kernel of \epsilon......Page 205 3. The rational canonical form......Page 207 4. The primary rational and Jordan canonical forms......Page 210 References......Page 215 Index......Page 217 An account of how a certain fundamental algebraic concept can be introduced, developed, and applied to solve some concrete algebraic problems.