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108 9. PRIMARY DECOMPOSITION and isomorphisms M Proof. This follows from 7.4.5. i MPi M/Pi niM 9.4.7. Proposition. Let R be a noetherian ring and M a finite module. Let L ⊂ M have a reduced primary decomposition L = N1 ∩ · · · ∩ Nn where Ni is Pi-primary. Assume U to be a multiplicative subset disjoint from exactly P1, . . . Pk. Then is a reduced primary decomposition. U −1 L = U −1 N1 ∩ · · · ∩ U −1 Nk Proof. This follows from 9.3.7 and 9.4.3. 9.4.8. Exercise. (1) Describe the primary decomposition over a field. 9.5. Decomposition of ideals 9.5.1. Proposition. A proper ideal I ⊂ R has a reduced primary decomposition and for any such and is exact. Proof. This is a case of 9.4.3. I = Q1 ∩ · · · ∩ Qn Ass(R/I) = {P1, . . . , Pn} 0 → R/I → R/Qi 9.5.2. Proposition. If I = Q1 ∩ · · · ∩ Qn is a reduced primary decomposition and Pi is minimal in Ass(R/I), then and therefore uniquely determined. Proof. This is a case of 9.4.4. Qi = R ∩ IRPi 9.5.3. Definition. Let P be a prime ideal. The symbolic power of P is P (n) = R ∩ P n RP 9.5.4. Proposition. Let P n = Q1 ∩ · · · ∩ Qn be a reduced primary decomposition of a power of a prime ideal P . If Ass(R/Q1) = {P }, then Q1 = P (n) Proof. This is a case of 9.4.4. i i
9.5. DECOMPOSITION OF IDEALS 109 9.5.5. Proposition. Let I ⊂ R be a proper ideal in a noetherian ring such that R/I has finite length. If Ass(R/I) = {P1, . . . , Pn}, then there is a reduced primary decomposition I = Q1 ∩ · · · ∩ Qn where and an isomorphism Proof. This is a case of 9.4.3 and 9.5.4. Qi = R ∩ QPi R/I R/Qi 9.5.6. Proposition. Let R be an artinian ring. If Ass(M) = {P1, . . . , Pn}, then there is a reduced primary decomposition where and isomorphisms Proof. This is a case of 9.4.6. 0 = Q1 ∩ · · · ∩ Qn Qi = Pi ni , R/Qi RPi R i RPi i R/Pi ni 9.5.7. Proposition. Let a proper ideal I ⊂ R in a noetherian ring have a reduced primary decomposition I = Q1 ∩ · · · ∩ Qn where Qi is Pi-primary. Assume U to be a multiplicative subset disjoint from exactly P1, . . . Pk. Then is a reduced primary decomposition. Proof. This is a case of 9.4.7. U −1 I = U −1 Q1 ∩ · · · ∩ U −1 Qk 9.5.8. Example. Let R be a unique factorization domain. A factorization into powers of different irreducible primes is a reduced primary decomposition of a principal ideal. 9.5.9. Proposition. Let R be a noetherian ring. The following are equivalent (1) R is reduced. (2) RP is a field for all P ∈ Ass(R). (3) RP is a domain for all P ∈ Ass(R). Proof. (1) ⇒ (2): The maximal ideal P RP = 0. (3) ⇒ (1): This follows from √ 0 = ∩P ∈Ass(R)P . 9.5.10. Corollary. Let R be a reduced noetherian ring. Then all elements in Ass(R) are minimal primes. That is, there are no embedded primes. 9.5.11. Exercise. (1) Let I ⊂ R be an ideal. Show that if P = √ I is a maximal ideal, then I is a P -primary ideal. (2) Let I ⊂ R be an ideal. Show that if I contains a power of a maximal ideal P , then I is a P -primary ideal. i
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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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Page 58 and 59: 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 78 and 79: 78 6. FINITE MODULES Let the n × n
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Page 94 and 95: 94 8. NOETHERIAN MODULES Proof. Use
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