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56 3. EXACT SEQUENCES OF MODULES Proof. Let E be an injective module. By 3.7.9 and 3.6.3 the sequence 0 HomR(F, E) is split exact. So also the sequence 0 HomR(L, HomR(F, E)) HomR(N, E) HomR(M, E) HomR(L, HomR(N, E)) HomR(L, HomR(M, E)) is split exact. By 2.6.13 this is natural isomorphic to the sequence 0 Conclusion by 3.6.13. HomR(L ⊗R F, E) HomR(L ⊗R N, E) HomR(L ⊗R M, E) 3.7.12. Proposition. A module F is flat, if for any ideal I ⊂ R is exact. 0 → I ⊗R F → R ⊗R F Proof. By 3.7.9 it suffices to see that HomR(F, E) is injective for any injective E. By 3.6.7 this amounts to HomR(R, HomR(F, E)) → HomR(I, HomR(F, E)) being surjective. By 2.6.13 this homomorphism is HomR(R⊗RF, E) → HomR(I⊗R F, E) which is surjective since E is injective. 3.7.13. Exercise. (1) Show that Z/(n) is not a flat Z-module for n = 0, 1. (2) Show that Z/(2) is a flat Z/(6)-module. (3) Show that Z/(2) is not a flat Z/(4)-module. 0 0 0
4 Fraction constructions 4.1. Rings of fractions 4.1.1. Lemma. Let R be a ring and U ⊂ R such that 1 ∈ U and for u, v ∈ U the product uv ∈ U. On U × R is defined a relation (u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′ (1) The relation is an equivalence relation. (2) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, va + ub) ∼ (u ′ v ′ , v ′ a ′ + u ′ b ′ ). (3) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, ab) ∼ (u ′ v ′ , a ′ b ′ ). Proof. The claims are proved by simple calculations. (1) Symmetry is clear. Reflexive follows as 1 ∈ U. Transitive: if (u, a) ∼ (u ′ , a ′ ), (u ′ , a ′ ) ∼ (u ′′ , a ′′ ) then vu ′ a = vua ′ , v ′ u ′′ a ′ = v ′ u ′ a ′′ . Since multiplication is commutative (v ′ vu ′ )u ′′ a = v ′ u ′′ vua ′ = (v ′ vu ′ )ua ′′ give (u, a) ∼ (u ′′ , a ′′ ). (2) From wu ′ a = wua ′ , w ′ v ′ b = w ′ vb ′ follow that w ′ vv ′ wu ′ a = w ′ vv ′ wua ′ , wuu ′ w ′ v ′ b = wuu ′ w ′ vb ′ . So now ww ′ (u ′ v ′ )(va + ub) = ww ′ (uv)(v ′ a ′ + u ′ b ′ ) as needed. (3) Is similar. 4.1.2. Definition. Let R be a ring. U ⊂ R is a multiplicative subset if 1 ∈ U and for u, v ∈ U the product uv ∈ U. The ring of fractions U −1R is given by on U × R under the relation 4.1.1 equivalence classes a u (u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′ The addition is a b + u v and the multiplication is a b · u v The canonical ring homomorphism is = va + ub uv = ab uv ι : R → U −1 R, a ↦→ a 1 4.1.3. Proposition. Let φ : R → S be a ring homomorphism and U ⊂ R a multiplicative subset. If all elements in φ(U) ⊂ S are units, then there exists a unique ring homomorphism φ ′ : U −1 R → S such that φ = φ ′ ◦ ι. φ R ι U −1R Proof. φ ′ ( a u ) = φ(a)φ(u)−1 is the well defined unique ring homomorphism. Observe that the elements of form bv −1 satisfy the rules for fractions. 57 φ ′ S
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ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5 and 6: Contents Prerequisites 7 1. A dicti
Page 7: Prerequisites The basic notions fro
Page 10 and 11: 10 1. A DICTIONARY ON RINGS AND IDE
Page 22 and 23: 22 2. MODULES (5) If R is a field,
Page 24 and 25: 24 2. MODULES (1) IM = {a1y1 + ·
Page 26 and 27: 26 2. MODULES 2.3.6. Corollary. Let
Page 28 and 29: 28 2. MODULES (2) Give an example o
Page 30 and 31: 30 2. MODULES 2.4.11. Proposition.
Page 32 and 33: 32 2. MODULES Proof. By 2.5.3 the c
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Page 36 and 37: 36 2. MODULES 2.6.12. Example. Let
Page 38 and 39: 38 2. MODULES (1) The change of rin
Page 40 and 41: 40 3. EXACT SEQUENCES OF MODULES 3.
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Page 72 and 73: 72 5. LOCALIZATION (1) φ is flat.
Page 74 and 75: 74 6. FINITE MODULES 6.1.7. Corolla
Page 76 and 77: 76 6. FINITE MODULES (4) Let A be a
Page 78 and 79: 78 6. FINITE MODULES Let the n × n
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Page 82 and 83: 82 6. FINITE MODULES 6.5.9. Corolla
Page 84 and 85: 84 6. FINITE MODULES (1) E is an in
Page 86 and 87: 86 7. MODULES OF FINITE LENGTH 7.2.
Page 88 and 89: 88 7. MODULES OF FINITE LENGTH Proo
Page 90 and 91: 90 7. MODULES OF FINITE LENGTH (5)
Page 92 and 93: 92 7. MODULES OF FINITE LENGTH 7.5.
Page 94 and 95: 94 8. NOETHERIAN MODULES Proof. Use
Page 96 and 97: 96 8. NOETHERIAN MODULES 8.3.2. Cor
Page 98 and 99: 98 8. NOETHERIAN MODULES 8.5. Fract
Page 101 and 102: 9 Primary decomposition 9.1. Suppor
Page 103 and 104: 9.2. ASS OF MODULES 103 Proof. The
Page 105 and 106: 9.2. ASS OF MODULES 105 9.2.12. Def
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9.4. DECOMPOSITION OF MODULES 107 9
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9.5. DECOMPOSITION OF IDEALS 109 9.
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10 Dedekind rings 10.1. Principal i
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10.3. DEDEKIND DOMAINS 113 10.2.4.
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10.3. DEDEKIND DOMAINS 115 10.3.10.
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118 BIBLIOGRAPHY
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120 INDEX finite type rings, 95 fin
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