Abstract Algebra (Holden-Day Series in Mathematics) by Andrew O Lindstrum

By Andrew O Lindstrum

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Preface ix
Chapter I units AND MAPS 3
I. units and Subsets 3
2. Maps S
3. Isomorphisms of units 7
4. Epimorphisms and Monomorphisms 8
S. the picture research of a Map 10
6. The Coimage research of a Map II
7. Description of Surjective Maps 12
8. Equivalence kinfolk 13
9. Cardinality of units IS
10. Ordered units 16
II. Axiom of selection 17
12. items and Sums of units 20
Exercises 23
Chapter 2 MONOIDS AND teams 27
1. Monoids 27
2. Morphisms of Monoids 30
3. precise varieties of Morphisms 32
4. Analyses of Morphisms 37
5. Description of Surjective Morphisms 39
6. teams and Morphisms of teams 41
7. Kernels of Morphisms of teams 43
8. teams of Fractions 49
9. The Integers 55
10. Finite and endless units 57
Exercises 64
Chapter three different types 75
1. different types 75
2. Morphisms 79
3. items and Sums 82
Exercises 85
Chapter four earrings 99
1. type of earrings 99
2. Polynomial earrings 103
3. Analyses of Ring Morphisms 107
4. beliefs 112
5. items of jewelry 115
Exercises 116
PART 127
Chapter five designated FACTORIZATION domain names 129
I. Divisibility 130
2. essential domain names 133
3. certain Factorization domain names 138
4. Divisibility in UFD\'s 140
5. central perfect domain names 147
6. issue jewelry of PID\'s 152
7. Divisors 155
8. Localization in crucial domain names 159
9. A Criterion for designated Factorization 164
10. while R [X] is a UFD 169
Exercises 171
Chapter 6 normal MODULE concept 176
1. classification of Modules over a hoop 178
2. The Composition Maps in Mod(R) 183
3. Analyses of R-Module Morphisms 185
4. unique Sequences 193
5. Isomorphism Theorems 201
6. Noetherian and Artinian Modules 206
7. unfastened R-Modules 210
8. Characterization of department earrings 216
9. Rank of loose Modules 221
10. Complementary Submodules of a Module 224
11. Sums of Modules 231
12. switch of jewelry 239
13. Torsion Modules over PID\'s 242
14. items of Modules 246
Exercises 248
Chapter 7 SEMISIMPLE jewelry AND MODULES 266
I. easy jewelry 266
2. Semisimple Modules 271
3. Projective Modules 276
4. the other Ring 280
Exercises 283
Chapter eight ARTINIAN jewelry 289
1. Idempotents in Left Artinian earrings 289
2. the unconventional of a Left Artinian Ring 294
3. the novel of an Arbitrary Ring 298
Exercises 302
PART 3 311
Chapter nine LOCALIZATION AND TENSOR items 313
1. Localization of jewelry 313
2. Localization of Modules 316
3. purposes of Localization 320
4. Tensor items 323
5. Morphisms of Tensor items 328
6. in the neighborhood unfastened Modules 334
Exercises 337
Chapter 10 valuable excellent domain names 351
I. Submodules of unfastened Modules 352
2. loose Submodules of loose Modules 355
3. Finitely Generated Modules over PID\'s 359
4. Injective Modules 363
5. the basic Theorem for PID\'s 366
Exercises 371
Chapter II purposes OF basic THEOREM 376
I. Diagonalization 376
2. Determinants 380
3. Mat rices 387
4. extra purposes of the basic Theorem 391
5. Canonical kinds 395
Exercises forty I
PART 4 413
1. Roots of Polynomials 415
2. Algebraic parts 420
3. Morphisms of Fields 425
4. Separability 430
5. Galois Extensions 434
Exercises 440
Chapter thirteen DEDEKIND domain names 445
I. Dedekind domain names 445
2. imperative Extensions 449
3. Characterizations of Dedekind domain names 454
4. beliefs 457
5. Finitely Generated Modules over Dedekind domain names 462
Exercises 463
Index 469

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Example text

Let X and Y be compact topological spaces with Y a metric space with distance function d. Let U be a set of continuous maps X −→ Y such that for every x ∈ X and every > 0 there exists a neighborhood N of x such that d f (x), f (x ) < for all x ∈ N and for all f ∈ U . Then every sequence in U has a uniformly convergent subsequence. We refer to the hypothesis on U as equicontinuity. Proof. Let S0 = {f1 , f2 , f3 , . } be a sequence in U . We will show that it has a convergent subsequence. We will construct a subsequence that is uniformly Cauchy and hence has a limit.

Deduce that k is odd. Hint: Normalize the vector field so that Xm is a unit tangent vector for all m. If m ∈ S k consider the great circle θm : [0, 2π] −→ S k tangent to Xm . Then θm (0) = θm (2π) = m, but m −→ θm (π) is the antipodal map. Also, think about the effect of the antipodal map on H k (S k ). 7 Left-Invariant Vector Fields To recapitulate, a Lie group is a differentiable manifold with a group structure in which the multiplication and inversion maps G × G −→ G and G −→ G are smooth. A homomorphism of Lie groups is a group homomorphism that is also a smooth map.

Xi Exercises The following exercise requires some knowledge of topology. 1. Let X be a vector field on the sphere S k . , Xm = 0 for all m ∈ S k . Show that the antipodal map a : S k −→ S k and the identity map S k −→ S k are homotopic. Deduce that k is odd. Hint: Normalize the vector field so that Xm is a unit tangent vector for all m. If m ∈ S k consider the great circle θm : [0, 2π] −→ S k tangent to Xm . Then θm (0) = θm (2π) = m, but m −→ θm (π) is the antipodal map. Also, think about the effect of the antipodal map on H k (S k ).

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