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

96 8. NOETHERIAN MODULES
8.3.2. Corollary. Let R be a noetherian ring and R → S a finite type ring over R.
Then S is noetherian.
8.3.3. Corollary. Let R be a noetherian ring and M a finite module. Then the ring
R ⊕ M, 2.1.14, is noetherian.
8.3.4. Example. Let I ⊂ R be an ideal in a noetherian ring. Then there are noetherian
rings.
(1) GIR = ⊕n≥0I n /I n+1 .
(2) BIR = ⊕n≥0I n .
8.3.5. Theorem (Krull’s intersection theorem). Let I be an ideal in a noetherian
ring R. Then there is a ∈ I such that
1 + a ∈ Ann(
I n )
Proof. Let I = (u1, . . . , um). If b ∈ I n then b = fn(u1, . . . , um) where fn ∈
R[X1, . . . , Xm] are homogeneous of degree n. By Hilbert’s basis theorem, 8.3.1
there is N such that
fN+1 = f1g1 + · · · + fNgN
where gn is homogeneous of degree N − n + 1 > 0. By substitution b ∈ bI and
I ∩ I n = ∩I n . Conclusion by 6.3.4.
8.3.6. Corollary. Let I be an ideal in a noetherian ring R such that the elements
1 + a, a ∈ I are nonzero divisors. Then
I n = 0
n
8.3.7. Corollary. Let I be an ideal in a noetherian ring R and M a finite module.
Then there is a ∈ I such that
1 + a ∈ Ann(
I n M)
Proof. Use 8.3.5 on the ring R ⊕ M and the ideal I ⊕ M. Clearly (I + M) n =
I n + I n−1 M.
8.3.8. Corollary. Let I be an ideal in a noetherian ring R and M a finite module
such that the elements 1 + a, a ∈ I are nonzero divisors on M. Then
I n M = 0
n
8.3.9. Proposition. If R ⊂ S be a finite extension. Then R is noetherian if and
only if S is noetherian.
Proof. Assume S is noetherian. Let Eα be a family of injective R-modules. Then
HomR(S, Eα) is an injective S-module. Since S is finite over R, HomR(S, Eα)
HomR(S, Eα) 6.1.14. By 6.6.7 Eα is injective and by 8.2.11 R is noetherian.
8.3.10. Example. Let K be a field and R = K[X1, X2, . . . ]/(X1 − x2X3, x2 −
x3X4, . . . ). Put P = (X1, X2, . . . ).
(1) P is maximal
(2) P 2 = P .
(3) P (∩nP n ) = ∩nP n .
n
n