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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10<br />
Dedekind rings<br />
10.1. Principal ideal domains<br />
10.1.1. Lemma. Let R be a domain. The set <strong>of</strong> elements x ∈ M in a module with<br />
Ann(x) = 0 is a submodule.<br />
Pro<strong>of</strong>. The product <strong>of</strong> two nonzero elements is nonzero.<br />
10.1.2. Definition. Let R be a domain. An element x ∈ M in a module is a torsion<br />
element if Ann(x) = 0. By 10.1.1 the set <strong>of</strong> torsion elements is a submodule<br />
T (M) ⊂ M, the torsion submodule. If T (M) = 0 then M is a torsion free<br />
module.<br />
10.1.3. Lemma. Let R be a domain.<br />
(1) A flat module is torsion free.<br />
(2) M/T (M) is torsion free.<br />
(3) Let K be the fraction field. Then T (M) = Ker(M → M ⊗R K).<br />
(4) If M → N → L is exact, then T (M) → T (N) → T (L) is exact.<br />
(5) If U ⊂ R is multiplicative, then U −1 T (M) = T (U −1 M).<br />
Pro<strong>of</strong>. (1) If 0 = a ∈ R and F flat, then aF is injective. (2), (4), (5) These are<br />
clear. (3) This follows from 4.4.1.<br />
10.1.4. Corollary. Let R be a domain and M a module. The following are equivalent.<br />
(1) M is torsion free.<br />
(2) MP is torsion free for all prime ideals P .<br />
(3) MP is torsion free for all maximal ideals P .<br />
10.1.5. Lemma. Let R be a principal ideal domain. A submodule <strong>of</strong> a finite free<br />
module is free.<br />
Pro<strong>of</strong>. Let F ⊂ R n be a submodule and p : R n → R the last projection. Then<br />
p(F ) is a principal ideal and free. By induction F ∩ Ker p is free, so F F ∩<br />
Ker p ⊕ p(F ) is free.<br />
10.1.6. Proposition. Let R be a principal ideal domain.<br />
(1) A torsion free module is flat.<br />
(2) A finite torsion free module is free.<br />
Pro<strong>of</strong>. (1) For a nonzero ideal (a) ⊂ R the composite R (a) → R is aM.<br />
For a torsion free F the homomorphism aF : F (a) ⊗R F → F is injective.<br />
So F is flat by 3.7.12. (2) Let K be the fraction field. Let F ⊂ F ⊗R K be a<br />
torsion free submodule and suppose x1, . . . , xn ∈ F give a basis for F ⊗R K.<br />
Then F ′ = Rx1 ⊕ · · · ⊕ Rxn ⊂ F is a free submodule such that F/F ′ is a finite<br />
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