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

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.
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