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

112 10. DEDEKIND RINGS
torsion module. There is 0 = a ∈ Ann(F/F ′ ), so F aF ⊂ F ′ . Conclusion by
10.1.5.
10.1.7. Proposition. Let R be a principal ideal domain. A finite module M decomposes
M = T (M) ⊕ F
as a direct sum of the torsion submodule and a finite free submodule F .
Proof. The sequence 0 → T (M) → M → M/T (M) → 0 is split exact.
10.1.8. Proposition. Let R be a principal ideal domain. A finite torsion module
M has a primary decomposition
Where
M = ⊕ (p)Mp
Mp = {x ∈ M|p n x = 0, for some n}
Proof. This is the primary decomposition 9.4.6.
10.1.9. Proposition. Let R be a principal ideal domain and (p) an irreducible
principal ideal. A finite torsion module M such that M = Mp has decomposition
Where n1 ≥ · · · ≥ nk.
M = R/(p n1 ) ⊕ · · · ⊕ R/(p nk )
Proof. Let n1 = n be such that p n ∈ Ann(M), but p n−1 x = 0 for some x ∈ M.
The short exact sequence 0 → Rx → M → M/Rx → 0 of R/(p n )-modules
is split exact, since R/(p n ) Rx is an injective module 7.5.9. Conclusion by
induction on ℓR(M).
10.1.10. Exercise. (1) Show that a nonzero finite Z-submodule of Q is a free module
of rank 1.
(2) Show that a finite torsion Z-module is a finite group.
(3) Let K be a field. Show that a finite torsion K[X]-module is a finite K vector space.
(4) Let R be a noetherian domain such that every nonzero prime ideal is maximal. Let
M be a finite module. Show that the torsion submodule T (M) has finite length.
10.2. Discrete valuation rings
10.2.1. Definition. A local principal ideal domain, which is not a field, is a discrete
valuation ring. A generator of the maximal ideal is a local parameter or a
uniformizing parameter.
10.2.2. Proposition. Let (R, (p)) be a discrete valuation ring. Then
Proof. See 8.5.5.
∩n(p n ) = 0
10.2.3. Proposition. Let (R, (p)) be a discrete valuation ring. Any nonzero ideal
is of the form (p n ) for a unique n = 0, 1, 2, . . . .
Proof. Let 0 = x R, by 10.2.2 there is n such that x ∈ (p n ) − (p n+1 ). Since any
ideal is finitely generated it follows that a nonzero ideal is of the form (p n ). n is
unique by unique factorization.