Documentation

Mathlib.RingTheory.DedekindDomain.Ideal.Basic

Dedekind domains and invertible ideals #

In this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible, and prove instances such as the unique factorization of ideals. Further results on the structure of ideals in a Dedekind domain are found in Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean.

Main definitions #

isDedekindDomain_iff_mul_inv_cancel shows an integral domain is a Dedekind domain iff every nonzero fractional ideal is invertible.

Main results: #

Implementation notes #

The definitions that involve a field of fractions choose a canonical field of fractions, but are independent of that choice. The ..._iff lemmas express this independence.

Often, definitions assume that Dedekind domains are not fields. We found it more practical to add a (h : ¬ IsField A) assumption whenever this is explicitly needed.

References #

Tags #

dedekind domain, dedekind ring

theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDomain A] (x : K) (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral (nonZeroDivisors A) x hx)) :

adjoinIntegral (nonZeroDivisors A) x hx = 1

theorem FractionalIdeal.one_mem_inv_coe_ideal {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDomain A] {I : Ideal A} (hI : I ≠ ⊥) :

1 ∈ (↑I)⁻¹

@[deprecated FractionalIdeal.one_mem_inv_coe_ideal (since := "2026-04-16")]

theorem one_mem_inv_coe_ideal {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDomain A] {I : Ideal A} (hI : I ≠ ⊥) :

1 ∈ (↑I)⁻¹

Alias of FractionalIdeal.one_mem_inv_coe_ideal.

theorem PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_le {A : Type u_2} [CommRing A] [IsDedekindDomain A] (hNF : ¬IsField A) {I M : Ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.IsMaximal] :

∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I

Specialization of exists_primeSpectrum_prod_le_and_ne_bot_of_domain to Dedekind domains: Let I : Ideal A be a nonzero ideal, where A is a Dedekind domain that is not a field. Then exists_primeSpectrum_prod_le_and_ne_bot_of_domain states we can find a product of prime ideals that is contained within I. This lemma extends that result by making the product minimal: let M be a maximal ideal that contains I, then the product including M is contained within I and the product excluding M is not contained within I.

@[deprecated PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_le (since := "2026-04-16")]

theorem exists_multiset_prod_cons_le_and_prod_not_le {A : Type u_2} [CommRing A] [IsDedekindDomain A] (hNF : ¬IsField A) {I M : Ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.IsMaximal] :

∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z).prod ≤ I ∧ ¬(Multiset.map PrimeSpectrum.asIdeal Z).prod ≤ I

Alias of PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_le.

Specialization of exists_primeSpectrum_prod_le_and_ne_bot_of_domain to Dedekind domains: Let I : Ideal A be a nonzero ideal, where A is a Dedekind domain that is not a field. Then exists_primeSpectrum_prod_le_and_ne_bot_of_domain states we can find a product of prime ideals that is contained within I. This lemma extends that result by making the product minimal: let M be a maximal ideal that contains I, then the product including M is contained within I and the product excluding M is not contained within I.

theorem FractionalIdeal.not_inv_le_one_of_ne_bot {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :

¬(↑I)⁻¹ ≤ 1

theorem FractionalIdeal.mul_inv_cancel_of_le_one {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥) (hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1) :

↑I * (↑I)⁻¹ = 1

theorem FractionalIdeal.coe_ideal_mul_inv {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :

↑I * (↑I)⁻¹ = 1

Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible, and finally conclude that fractional ideals in a Dedekind domain form a group with zero.

theorem isDedekindDomain_iff_mul_inv_cancel {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] [IsDomain A] :

IsDedekindDomain A ↔ ∀ (I : FractionalIdeal (nonZeroDivisors A) K), I ≠ ⊥ → I * I⁻¹ = 1

@[instance_reducible]

noncomputable instance FractionalIdeal.semifield {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] :

Semifield (FractionalIdeal (nonZeroDivisors A) K)

Equations

One or more equations did not get rendered due to their size.

instance FractionalIdeal.instPosMulStrictMonoNonZeroDivisors {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] :

PosMulStrictMono (FractionalIdeal (nonZeroDivisors A) K)

instance FractionalIdeal.instMulPosStrictMonoNonZeroDivisors {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] :

MulPosStrictMono (FractionalIdeal (nonZeroDivisors A) K)

instance FractionalIdeal.instPosMulReflectLENonZeroDivisors {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] :

PosMulReflectLE (FractionalIdeal (nonZeroDivisors A) K)

instance FractionalIdeal.instMulPosReflectLENonZeroDivisors {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] :

MulPosReflectLE (FractionalIdeal (nonZeroDivisors A) K)

theorem FractionalIdeal.mul_left_strictMono {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] {I : FractionalIdeal (nonZeroDivisors A) K} (hI : I ≠ 0) :

StrictMono fun (x : FractionalIdeal (nonZeroDivisors A) K) => x * I

theorem FractionalIdeal.mul_right_strictMono {A : Type u_2} (K : Type u_3) [CommRing A] [Field K] [IsDedekindDomain A] [Algebra A K] [IsFractionRing A K] {I : FractionalIdeal (nonZeroDivisors A) K} (hI : I ≠ 0) :

StrictMono fun (x : FractionalIdeal (nonZeroDivisors A) K) => I * x

instance FractionalIdeal.instPosMulReflectLEIdeal {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

PosMulReflectLE (Ideal A)

instance Ideal.isCancelMulZero {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

IsCancelMulZero (Ideal A)

instance Ideal.isDomain {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

IsDomain (Ideal A)

instance instPosMulStrictMonoIdeal {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

PosMulStrictMono (Ideal A)

instance instMulPosStrictMonoIdeal {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

MulPosStrictMono (Ideal A)

theorem Ideal.dvd_iff_le {A : Type u_2} [CommRing A] [IsDedekindDomain A] {I J : Ideal A} :

I ∣ J ↔ J ≤ I

For ideals in a Dedekind domain, to divide is to contain.

theorem Ideal.liesOver_iff_dvd_map {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [IsDedekindDomain A] [Algebra R A] {p : Ideal R} {P : Ideal A} (hP : P ≠ ⊤) [p.IsMaximal] :

P.LiesOver p ↔ P ∣ map (algebraMap R A) p

theorem Ideal.dvdNotUnit_iff_lt {A : Type u_2} [CommRing A] [IsDedekindDomain A] {I J : Ideal A} :

DvdNotUnit I J ↔ J < I

instance instWfDvdMonoidIdeal {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

WfDvdMonoid (Ideal A)

instance Ideal.uniqueFactorizationMonoid {A : Type u_2} [CommRing A] [IsDedekindDomain A] :

UniqueFactorizationMonoid (Ideal A)

@[instance_reducible]

noncomputable instance Ideal.strongNormalizationMonoid {A : Type u_2} [CommRing A] :

StrongNormalizationMonoid (Ideal A)

Equations

Ideal.strongNormalizationMonoid = inferInstance

@[deprecated Ideal.strongNormalizationMonoid (since := "2026-07-08")]

def Ideal.normalizationMonoid {A : Type u_2} [CommRing A] :

StrongNormalizationMonoid (Ideal A)

Alias of Ideal.strongNormalizationMonoid.

Equations

@Ideal.normalizationMonoid = @Ideal.strongNormalizationMonoid

Instances For