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Mathlib.Algebra.Module.DedekindDomain

Modules over a Dedekind domain #

Over a Dedekind domain, an I-torsion module is the internal direct sum of its p i ^ e i-torsion submodules, where I = ∏ i, p i ^ e i is its unique decomposition in prime ideals. Therefore, as any finitely generated torsion module is I-torsion for some I, it is an internal direct sum of its p i ^ e i-torsion submodules for some prime ideals p i and numbers e i.

Over a Dedekind domain, an I-torsion module is the internal direct sum of its p i ^ e i- torsion submodules, where I = ∏ i, p i ^ e i is its unique decomposition in prime ideals.

A finitely generated torsion module over a Dedekind domain is an internal direct sum of its p i ^ e i-torsion submodules where p i are factors of (⊤ : Submodule R M).annihilator and e i are their multiplicities.

theorem Submodule.exists_isInternal_prime_power_torsion {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M] [IsDedekindDomain R] [Module.Finite R M] (hM : Module.IsTorsion R M) :
∃ (P : Finset (Ideal R)) (x : DecidableEq { x : Ideal R // x P }) (_ : pP, Prime p) (e : { x : Ideal R // x P }), DirectSum.IsInternal fun (p : { x : Ideal R // x P }) => Submodule.torsionBySet R M (p ^ e p)

A finitely generated torsion module over a Dedekind domain is an internal direct sum of its p i ^ e i-torsion submodules for some prime ideals p i and numbers e i.