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

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110 9. PRIMARY DECOMPOSITION (3) Let K be a field and I = (X 2 , XY ) ⊂ K[X, Y ]. Show that √ I = (X), but I is not (X)-primary.
10 Dedekind rings 10.1. Principal ideal domains 10.1.1. Lemma. Let R be a domain. The set of elements x ∈ M in a module with Ann(x) = 0 is a submodule. Proof. The product of two nonzero elements is nonzero. 10.1.2. Definition. Let R be a domain. An element x ∈ M in a module is a torsion element if Ann(x) = 0. By 10.1.1 the set of torsion elements is a submodule T (M) ⊂ M, the torsion submodule. If T (M) = 0 then M is a torsion free module. 10.1.3. Lemma. Let R be a domain. (1) A flat module is torsion free. (2) M/T (M) is torsion free. (3) Let K be the fraction field. Then T (M) = Ker(M → M ⊗R K). (4) If M → N → L is exact, then T (M) → T (N) → T (L) is exact. (5) If U ⊂ R is multiplicative, then U −1 T (M) = T (U −1 M). Proof. (1) If 0 = a ∈ R and F flat, then aF is injective. (2), (4), (5) These are clear. (3) This follows from 4.4.1. 10.1.4. Corollary. Let R be a domain and M a module. The following are equivalent. (1) M is torsion free. (2) MP is torsion free for all prime ideals P . (3) MP is torsion free for all maximal ideals P . 10.1.5. Lemma. Let R be a principal ideal domain. A submodule of a finite free module is free. Proof. Let F ⊂ R n be a submodule and p : R n → R the last projection. Then p(F ) is a principal ideal and free. By induction F ∩ Ker p is free, so F F ∩ Ker p ⊕ p(F ) is free. 10.1.6. Proposition. Let R be a principal ideal domain. (1) A torsion free module is flat. (2) A finite torsion free module is free. Proof. (1) For a nonzero ideal (a) ⊂ R the composite R (a) → R is aM. For a torsion free F the homomorphism aF : F (a) ⊗R F → F is injective. So F is flat by 3.7.12. (2) Let K be the fraction field. Let F ⊂ F ⊗R K be a torsion free submodule and suppose x1, . . . , xn ∈ F give a basis for F ⊗R K. Then F ′ = Rx1 ⊕ · · · ⊕ Rxn ⊂ F is a free submodule such that F/F ′ is a finite 111
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ELEMENTARY COMMUTATIVE ALGEBRA LECT
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Contents Prerequisites 7 1. A dicti
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Prerequisites The basic notions fro
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10 1. A DICTIONARY ON RINGS AND IDE
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22 2. MODULES (5) If R is a field,
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24 2. MODULES (1) IM = {a1y1 + ·
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26 2. MODULES 2.3.6. Corollary. Let
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28 2. MODULES (2) Give an example o
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30 2. MODULES 2.4.11. Proposition.
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32 2. MODULES Proof. By 2.5.3 the c
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34 2. MODULES (3) The formation of
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36 2. MODULES 2.6.12. Example. Let
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38 2. MODULES (1) The change of rin
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40 3. EXACT SEQUENCES OF MODULES 3.
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42 3. EXACT SEQUENCES OF MODULES Fo
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44 3. EXACT SEQUENCES OF MODULES an
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46 3. EXACT SEQUENCES OF MODULES Pr
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58 4. FRACTION CONSTRUCTIONS 4.1.4.
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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
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Page 94 and 95: 94 8. NOETHERIAN MODULES Proof. Use
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Commutative algebra - Department of Mathematical Sciences - old ...

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