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

98 8. NOETHERIAN MODULES
8.5. Fractions and localization
8.5.1. Proposition. Let R be a noetherian ring and U a multiplicative subset. Then
U −1 R is a noetherian ring.
Proof. Any ideal is extended 4.3.6.
8.5.2. Corollary. Let R be a noetherian ring and U a multiplicative subset. If M
is a noetherian R-module, then U −1 M is a noetherian U −1 R-module.
Proof. By 6.1.3 change of ring of a finite module is finite.
8.5.3. Corollary. Let R be a noetherian ring and M a finite module. Let P be a
prime ideal. Then RP is a noetherian ring and MP is a finite RP -module.
8.5.4. Proposition (Krull’s intersection theorem). Let (R, P ) be a noetherian local
ring and M a finite module. Then
P n M = 0
Proof. This follows from 8.3.7.
n
8.5.5. Corollary. Let (R, P ) be a noetherian local ring and I ⊂ P an ideal. Then
I n = 0
n
8.5.6. Proposition. Let (R, P ) be a noetherian local ring and F a finite module.
The following conditions are equivalent.
(1) F is free.
(2) F is projective.
(3) F is flat.
(4) P ⊗R F → F is injective.
Proof. This follows from 6.5.13.
8.5.7. Exercise. (1) Is it true that if U −1 R is noetherian, then R is noetherian?
8.6. Prime filtrations of modules
8.6.1. Proposition. Let R be a ring and M = 0 a nonzero module. An ideal
Ann(x) maximal in the set of ideals {Ann(y)|0 = y ∈ M} is a prime ideal.
Proof. Let Ann(x) be a maximal annihilator. Suppose a, b ∈ R such that ab ∈
Ann(x) and b /∈ Ann(x). Then
Ann(x) ⊆ Ann(bx) = R
Consequently Ann(x) = Ann(bx) in particular a ∈ Ann(x).
8.6.2. Corollary. Let R be a noetherian ring and M = 0 a nonzero module. Then
there is x ∈ M such that Ann(x) is a prime ideal.
8.6.3. Theorem. Let R be a noetherian ring and M = 0 a finite R-module. Then
there exists a finite filtration of M by submodules
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mr−1 ⊂ Mr = M
such that Mi/Mi−1, i = 1, . . . , r is isomorphic to an R-module of the form R/Pi
where Pi is a prime ideal in R.