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

5
Localization
5.1. Prime ideals
5.1.1. Theorem (Krull). A nonzero ring contains a maximal ideal.
Proof. The nonempty set of ideals different from R is ordered by inclusion. Given
an increasing chain Iα then ∪Iα is an ideal different from R which is a maximum
for the chain. Conclusion by Zorn’s lemma.
5.1.2. Corollary. Any proper ideal in a ring is contained in a maximal ideal.
Proof. The factor ring is nonzero and contains a maximal ideal. Conclusion by
1.2.10.
5.1.3. Proposition. Let P1, . . . , Pn ⊂ R be ideals with at most 2 not being prime
ideals. If an ideal
I ⊂ P1 ∪ · · · ∪ Pn
then I ⊆ Pi for some i.
Proof. Assume n > 1 and I is not contained in any sub union. Moreover assume
the numbering such that P3, . . . , Pn are prime ideals. Then for each i there is
The element
ai ∈ (I ∩ Pi)\ ∪j=i Pj
an + a1 . . . an−1
is in I but not in any Pi, giving a contradiction. So n = 1.
5.1.4. Proposition. Let P1, . . . , Pn ⊂ R be prime ideals and I any ideal. If for
some a ∈ R
a + I ⊂ P1 ∪ · · · ∪ Pn
then I ⊆ Pi for some i.
Proof. If a ∈ ∩iPi then conclusion by 5.1.3. On the contrary after renumbering
there exists j with 1 ≤ j < n such that
a ∈ P1 ∩ · · · ∩ Pj\Pj+1 ∪ · · · ∪ Pn
Assume no inclusions between the prime ideals and choose by 5.1.3
b ∈ I ∩ Pj+1 ∩ · · · ∩ Pn\P1 ∪ · · · ∪ Pj
Then a + b /∈ ∪iPi contradicts the hypothesis.
5.1.5. Proposition. Let R → U −1 R be the canonical homomorphism. Extension
and contraction gives a bijective correspondence between prime ideals in R disjoint
from U and all prime ideals in U −1 R.
(1) For a prime ideal P ⊂ R\U the extended ideal P U −1 R is a prime ideal in
U −1 R and the contracted P U −1 R ∩ R = P .
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