Commutative algebra - Department of Mathematical Sciences - old ...

Contents
Prerequisites 7
1. A dictionary on rings and ideals 9
1.1. Rings 9
1.2. Ideals 11
1.3. Prime ideals 13
1.4. Chinese remainders 14
1.5. Unique factorization 15
1.6. Polynomials 16
1.7. Roots 18
1.8. Fields 19
1.9. Power series 20
2. Modules 21
2.1. Modules and homomorphisms 21
2.2. Submodules and factor modules 23
2.3. Kernel and cokernel 25
2.4. Sum and product 28
2.5. Homomorphism modules 30
2.6. Tensor product modules 33
2.7. Change of rings 36
3. Exact sequences of modules 39
3.1. Exact sequences 39
3.2. The snake lemma 43
3.3. Exactness of Hom 48
3.4. Exactness of Tensor 49
3.5. Projective modules 50
3.6. Injective modules 52
3.7. Flat modules 54
4. Fraction constructions 57
4.1. Rings of fractions 57
4.2. Modules of fractions 58
4.3. Exactness of fractions 60
4.4. Tensor modules of fractions 62
4.5. Homomorphism modules of fractions 63
4.6. The polynomial ring is factorial 64
5. Localization 65
5.1. Prime ideals 65
5.2. Localization of rings 67
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