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106 9. PRIMARY DECOMPOSITION (4) Let I, J ⊂ R be a ideals such that JRP ⊂ IRP for all P ∈ Ass(R/I). Show that J ⊂ I. 9.3. Primary modules 9.3.1. Definition. A submodule N ⊂ M is a primary submodule or more precisely P -primary if Ass(M/N) = {P }. 9.3.2. Proposition. A prime ideal P ⊂ R is a P -primary submodule. Proof. This is 9.2.2. 9.3.3. Proposition. Let R be a noetherian ring and M a finite module. For a submodule N ⊂ M the following are equivalent. (1) N ⊂ M is primary for some prime (2) The set of zero divisors on M/N is contained in the radical Ann(M/N). Proof. (1) ⇒ (2): Ann(M/N) = P the set of zero divisors by 9.2.9. (2) ⇒ (1): If P1, P2 ∈ Ass(M/N) then P1 ∪ P2 ⊂ Ann(M/N) ⊂ P1 ∩ P2, so P1 = P2. 9.3.4. Corollary. Let R be a noetherian ring and I ⊂ R a proper ideal. The following are equivalent. (1) I ⊂ R is primary for some prime. (2) Any zero divisor in R/I is nilpotent. 9.3.5. Corollary. Let R be a noetherian ring. (1) If an ideal I ⊂ R is P -primary then √ I = P . (2) If the radical √ I = P is a maximal ideal, then I ⊂ R is a P -primary submodule. (3) A finite power P n ⊂ R of a maximal ideal is a P -primary submodule. 9.3.6. Proposition. Let R be a noetherian ring and M a finite module. (1) If N ⊂ M is P -primary, then Ann(M/N) ⊂ R is P -primary. (2) If N, N ′ ⊂ M are P -primary, then N ∩ N ′ is P -primary. Proof. (1) This follows from 9.3.3. (2) This follows from 9.2.5. 9.3.7. Proposition. Let R be a noetherian ring and M a finite module. Suppose N ⊂ M is P -primary and U ⊂ R is multiplicative subset. (1) If U ∩ P = ∅, then U −1 N ⊂ U −1 M is P U −1 R-primary. (2) If U ∩ P = ∅, then U −1 N = U −1 M. Proof. This follows from 9.2.10. 9.3.8. Exercise. (1) Let K be a field. Show that (X 2 , Y ) ⊂ K[X, Y ] is (X, Y )primary. (2) Let p be a prime number. Show that (p k ) ⊂ Z is a primary ideal. 9.4. Decomposition of modules 9.4.1. Definition. A submodule L ⊂ M has a primary decomposition if there exist a family Ni ⊂ M of Pi-primary submodules, such that L = N1 ∩ · · · ∩ Nn A primary decomposition is a reduced primary decomposition if Pi = Pj for i = j and no Ni can be excluded.
9.4. DECOMPOSITION OF MODULES 107 9.4.2. Lemma. Let R be a noetherian ring and M a finite module. For each Pi ∈ Ass(M) there is a submodule Ni ⊆ M such that Ass(Ni) = Ass(M) − {Pi} and Ass(M/Ni)) = {Pi}. M injects 0 → M → M/Ni Proof. The submodule Ni is given by 9.2.6. Ass(∩Ni) = ∅, so conclusion by 9.2.2. 9.4.3. Proposition. Let R be a noetherian ring and M a finite module. A proper submodule L ⊂ M has a reduced primary decomposition and for any such and is exact. Proof. Apply 9.4.2 to M/L. L = N1 ∩ · · · ∩ Nn Ass(M/L) = {P1, . . . , Pn} 0 → M/L → M/Ni 9.4.4. Proposition. Let R be a noetherian ring and M a finite module. If L = N1 ∩ · · · ∩ Nn is a reduced primary decomposition of L ⊂ M and Pi is minimal in Ass(M/L), then and therefore uniquely determined. Ni = M ∩ LPi Proof. Clearly Ni ⊂ M ∩ LPi . By localization Ass(M ∩ LPi /Ni) = ∅. So equality. 9.4.5. Proposition. Let R be a noetherian ring and M a finite module. Let L ⊂ M such that M/L = 0 has finite length. If Ass(M/L) = {P1, . . . , Pn}, then there is a reduced primary decomposition where and an isomorphism L = N1 ∩ · · · ∩ Nn Ni = M ∩ LPi M/L M/Ni Proof. This follows from 9.4.3, 9.4.4 and 9.1.6. 9.4.6. Proposition. Let R be a noetherian ring and M a finite length module. If Ass(M) = {P1, . . . , Pn}, then there is a reduced primary decomposition where 0 = N1 ∩ · · · ∩ Nn Ni = Pi ni M, M/Ni MPi i i 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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54 3. EXACT SEQUENCES OF MODULES Pr
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