Commutative Algebra
نویسنده
چکیده
Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2.1. The basic formalism 20 2.2. Lattice Properties 22 2.3. Examples of Antitone Galois Connections 22 2.4. Antitone Galois Connections Decorticated: Relations 25 2.5. Isotone Galois Connections 26 2.6. Examples of Isotone Galois Connections 26 3. Modules 27 3.1. Basic definitions 27 3.2. Finitely presented modules 32 3.3. Torsion and torsionfree modules 34 3.4. Tensor and Hom 35 3.5. Projective modules 37 3.6. Injective modules 44 3.7. Flat modules 51 3.8. Nakayama’s Lemma 52 3.9. Ordinal Filtrations and Applications 56 3.10. Tor and Ext 64 3.11. More on flat modules 72 3.12. Faithful flatness 78 4. First Properties of Ideals in a Commutative Ring 81 4.1. Introducing maximal and prime ideals 81 4.2. Radicals 84
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