Ring Theory (eBook)

Front Cover 1
RING THEORY 4
Copyright Page 5
Contents 6
Preface 8
Chapter I. Basic Concepts 12
1 . Embedding a ring R in the ring of endomorphisms of an abelian group 12
2. R-Linear mappings of an R-module into itself 21
3. Vector spaces 24
4. Algebras 29
Chapter II. Primitive Rings 34
1. Dense rings of linear transformations of a vector space into itself 34
2. The finite topology 39
3. The lattice of left ideals in a primitive artinian ring 42
4. Homomorphisms of semigroups, rings, and modules relations
5. Simple rings with minimal left ideals 51
6. Isomorphism theorems 57
7. Modular maximal left ideals 61
8. Primitive algebras 65
Chapter III. Rings with a Faithful Family of Irreducible Modules 69
1. The radical of a ring 69
2. Semisimple rings as subdirect sums of primitive rings 74
3. Semisimple artinian rings 79
Chapter IV. Completely Reducible Modules 85
1. Direct sums of modules 86
2. Homogeneous components of an R-module 91
3. The double centralizer of a completely reducible module 95
4. Modules with boolean lattice of submodules 101
Chapter V. Tensor Products, Fields, and Matrix Representations 104
1. Polynomials 104
2. Fields 109
3. Tensor products 117
4. Representations by matrices 124
5. The trace of a separable field extension 132
Chapter VI. Separable Algebras 136
1. Central, simple algebras 136
2. The commutativity of rings satisfying xn(x) = x 147
3. The quarternions 152
4. The Wedderburn principal theorem 154
Chapter VII. Rings with Identity 159
1. Semiperfect rings 160
2. Projective modules and Asano orders 177
3. Injective modules and self-injective rings 185
Chapter VIII. Frobenius Algebras 196
1. The dual of an algebra module 197
2. Characterization of Frobenius algebras 200
3. Examples 210
4. Injective modules of a Frobenius algebra R 214
5. Behavior of semisimple algebras under separable extensions of the ground field 219
Chapter IX. Distributively Representable Rings 222
1. Modules with distributive lattice of submodules 222
2. Rings with distributive lattice of left ideals 233
3. Arithmetic rings 240
Chapter X. Noetherian Ideal Theory in Nonassociative Rings 260
Chapter XI. Orders in Semisimple Artinian Rings 271
Chapter XII. Rings of Continuous Functions 293
1. Biregular right algebras 293
2. The structure space of a ring 302
3. The theorem of Arens and Kaplansky 311
4. Boolean rings 317
Guide to the Literature 320
Bibliography 325
Index 328
Pure and Applied Mathematics: A Series of Monographs and Textbooks 332