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

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.