Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
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9.5. DECOMPOSITION OF IDEALS 109<br />
9.5.5. Proposition. Let I ⊂ R be a proper ideal in a noetherian ring such that R/I<br />
has finite length. If Ass(R/I) = {P1, . . . , Pn}, then there is a reduced primary<br />
decomposition<br />
I = Q1 ∩ · · · ∩ Qn<br />
where<br />
and an isomorphism<br />
Pro<strong>of</strong>. This is a case <strong>of</strong> 9.4.3 and 9.5.4.<br />
Qi = R ∩ QPi<br />
R/I <br />
R/Qi<br />
9.5.6. Proposition. Let R be an artinian ring. If Ass(M) = {P1, . . . , Pn}, then<br />
there is a reduced primary decomposition<br />
where<br />
and isomorphisms<br />
Pro<strong>of</strong>. This is a case <strong>of</strong> 9.4.6.<br />
0 = Q1 ∩ · · · ∩ Qn<br />
Qi = Pi ni , R/Qi RPi<br />
R <br />
i<br />
RPi<br />
i<br />
<br />
R/Pi ni<br />
9.5.7. Proposition. Let a proper ideal I ⊂ R in a noetherian ring have a reduced<br />
primary decomposition<br />
I = Q1 ∩ · · · ∩ Qn<br />
where Qi is Pi-primary. Assume U to be a multiplicative subset disjoint from<br />
exactly P1, . . . Pk. Then<br />
is a reduced primary decomposition.<br />
Pro<strong>of</strong>. This is a case <strong>of</strong> 9.4.7.<br />
U −1 I = U −1 Q1 ∩ · · · ∩ U −1 Qk<br />
9.5.8. Example. Let R be a unique factorization domain. A factorization into<br />
powers <strong>of</strong> different irreducible primes is a reduced primary decomposition <strong>of</strong> a<br />
principal ideal.<br />
9.5.9. Proposition. Let R be a noetherian ring. The following are equivalent<br />
(1) R is reduced.<br />
(2) RP is a field for all P ∈ Ass(R).<br />
(3) RP is a domain for all P ∈ Ass(R).<br />
Pro<strong>of</strong>. (1) ⇒ (2): The maximal ideal P RP = 0. (3) ⇒ (1): This follows from<br />
√ 0 = ∩P ∈Ass(R)P .<br />
9.5.10. Corollary. Let R be a reduced noetherian ring. Then all elements in<br />
Ass(R) are minimal primes. That is, there are no embedded primes.<br />
9.5.11. Exercise. (1) Let I ⊂ R be an ideal. Show that if P = √ I is a maximal ideal,<br />
then I is a P -primary ideal.<br />
(2) Let I ⊂ R be an ideal. Show that if I contains a power <strong>of</strong> a maximal ideal P , then I<br />
is a P -primary ideal.<br />
i