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ring_theory.dedekind_domain.ideal

Dedekind domains and ideals #

In this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible. Then we prove some results on the unique factorization monoid structure of the ideals.

Main definitions #

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 : ¬ is_field A) assumption whenever this is explicitly needed.

References #

Tags #

dedekind domain, dedekind ring

@[protected, instance]
noncomputable def fractional_ideal.has_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :
Equations
theorem fractional_ideal.inv_eq (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal (non_zero_divisors R₁) K} :
I⁻¹ = 1 / I
theorem fractional_ideal.inv_zero' (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :
0⁻¹ = 0
theorem fractional_ideal.inv_nonzero (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {J : fractional_ideal (non_zero_divisors R₁) K} (h : J 0) :
J⁻¹ = 1 / J, _⟩
theorem fractional_ideal.coe_inv_of_nonzero (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {J : fractional_ideal (non_zero_divisors R₁) K} (h : J 0) :
theorem fractional_ideal.mem_inv_iff {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal (non_zero_divisors R₁) K} (hI : I 0) {x : K} :
x I⁻¹ ∀ (y : K), y Ix * y 1
theorem fractional_ideal.inv_anti_mono {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I J : fractional_ideal (non_zero_divisors R₁) K} (hI : I 0) (hJ : J 0) (hIJ : I J) :
theorem fractional_ideal.le_self_mul_inv {K : Type u_3} [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal (non_zero_divisors R₁) K} (hI : I 1) :
I I * I⁻¹
theorem fractional_ideal.coe_ideal_le_self_mul_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : ideal R₁) :
theorem fractional_ideal.right_inverse_eq (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I J : fractional_ideal (non_zero_divisors R₁) K) (h : I * J = 1) :
J = I⁻¹

I⁻¹ is the inverse of I if I has an inverse.

theorem fractional_ideal.mul_inv_cancel_iff (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal (non_zero_divisors R₁) K} :
I * I⁻¹ = 1 ∃ (J : fractional_ideal (non_zero_divisors R₁) K), I * J = 1
theorem fractional_ideal.mul_inv_cancel_iff_is_unit (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {I : fractional_ideal (non_zero_divisors R₁) K} :
@[simp]
theorem fractional_ideal.map_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] {K' : Type u_5} [field K'] [algebra R₁ K'] [is_fraction_ring R₁ K'] (I : fractional_ideal (non_zero_divisors R₁) K) (h : K ≃ₐ[R₁] K') :
theorem fractional_ideal.invertible_of_principal (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : fractional_ideal (non_zero_divisors R₁) K) [I.is_principal] (h : I 0) :
I * I⁻¹ = 1
theorem fractional_ideal.is_principal_inv (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] (I : fractional_ideal (non_zero_divisors R₁) K) [I.is_principal] (h : I 0) :
@[simp]
theorem fractional_ideal.inv_one (K : Type u_3) [field K] {R₁ : Type u_4} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K] :
1⁻¹ = 1
def is_dedekind_domain_inv (A : Type u_2) [comm_ring A] [is_domain A] :
Prop

A Dedekind domain is an integral domain such that every fractional ideal has an inverse.

This is equivalent to is_dedekind_domain. In particular we provide a fractional_ideal.comm_group_with_zero instance, assuming is_dedekind_domain A, which implies is_dedekind_domain_inv. For integral ideals, is_dedekind_domain(_inv) implies only ideal.cancel_comm_monoid_with_zero.

Equations
theorem is_dedekind_domain_inv_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] :
theorem is_dedekind_domain_inv.mul_inv_eq_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal (non_zero_divisors A) K} (hI : I 0) :
I * I⁻¹ = 1
theorem is_dedekind_domain_inv.inv_mul_eq_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal (non_zero_divisors A) K} (hI : I 0) :
I⁻¹ * I = 1
@[protected]
theorem is_dedekind_domain_inv.is_unit {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A) {I : fractional_ideal (non_zero_divisors A) K} (hI : I 0) :

Showing one side of the equivalence between the definitions is_dedekind_domain_inv and is_dedekind_domain of Dedekind domains.

Specialization of exists_prime_spectrum_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_prime_spectrum_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 fractional_ideal.exists_not_mem_one_of_ne_bot {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] (hNF : ¬is_field A) {I : ideal A} (hI0 : I ) (hI1 : I ) :
∃ (x : K), x (I)⁻¹ x 1
theorem fractional_ideal.one_mem_inv_coe_ideal {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] {I : ideal A} (hI : I ) :
theorem fractional_ideal.mul_inv_cancel_of_le_one {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [h : is_dedekind_domain A] {I : ideal A} (hI0 : I ) (hI : (I * (I)⁻¹)⁻¹ 1) :
I * (I)⁻¹ = 1
theorem fractional_ideal.coe_ideal_mul_inv {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [h : is_dedekind_domain 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.

@[protected]
theorem fractional_ideal.mul_inv_cancel {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal (non_zero_divisors A) K} (hne : I 0) :
I * I⁻¹ = 1

Nonzero fractional ideals in a Dedekind domain are units.

This is also available as _root_.mul_inv_cancel, using the comm_group_with_zero instance defined below.

theorem fractional_ideal.mul_right_le_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {J : fractional_ideal (non_zero_divisors A) K} (hJ : J 0) {I I' : fractional_ideal (non_zero_divisors A) K} :
I * J I' * J I I'
theorem fractional_ideal.mul_left_le_iff {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {J : fractional_ideal (non_zero_divisors A) K} (hJ : J 0) {I I' : fractional_ideal (non_zero_divisors A) K} :
J * I J * I' I I'
theorem fractional_ideal.mul_right_strict_mono {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal (non_zero_divisors A) K} (hI : I 0) :
theorem fractional_ideal.mul_left_strict_mono {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] {I : fractional_ideal (non_zero_divisors A) K} (hI : I 0) :
@[protected]
theorem fractional_ideal.div_eq_mul_inv {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [algebra A K] [is_fraction_ring A K] [is_dedekind_domain A] (I J : fractional_ideal (non_zero_divisors A) K) :
I / J = I * J⁻¹

This is also available as _root_.div_eq_mul_inv, using the comm_group_with_zero instance defined below.

is_dedekind_domain and is_dedekind_domain_inv are equivalent ways to express that an integral domain is a Dedekind domain.

@[protected, instance]
Equations
theorem ideal.dvd_iff_le {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I J : ideal A} :
I J J I

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

theorem ideal.dvd_not_unit_iff_lt {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I J : ideal A} :
@[protected, instance]
@[simp]
theorem ideal.dvd_span_singleton {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {I : ideal A} {x : A} :
I ideal.span {x} x I
theorem ideal.is_prime_of_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (h : prime P) :
theorem ideal.prime_of_is_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (hP : P ) (h : P.is_prime) :
theorem ideal.prime_iff_is_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} (hP : P ) :

In a Dedekind domain, the (nonzero) prime elements of the monoid with zero ideal A are exactly the prime ideals.

theorem ideal.is_prime_iff_bot_or_prime {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] {P : ideal A} :

In a Dedekind domain, the the prime ideals are the zero ideal together with the prime elements of the monoid with zero ideal A.

theorem ideal.strict_anti_pow {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] (I : ideal A) (hI0 : I ) (hI1 : I ) :
theorem ideal.pow_lt_self {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] (I : ideal A) (hI0 : I ) (hI1 : I ) (e : ) (he : 2 e) :
I ^ e < I
theorem ideal.exists_mem_pow_not_mem_pow_succ {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] (I : ideal A) (hI0 : I ) (hI1 : I ) (e : ) :
∃ (x : A) (H : x I ^ e), x I ^ (e + 1)
theorem associates.le_singleton_iff {A : Type u_2} [comm_ring A] [is_domain A] [is_dedekind_domain A] (x : A) (n : ) (I : ideal A) :
theorem ideal.exist_integer_multiples_not_mem {A : Type u_2} {K : Type u_3} [comm_ring A] [field K] [is_domain A] [is_dedekind_domain A] [algebra A K] [is_fraction_ring A K] {J : ideal A} (hJ : J ) {ι : Type u_1} (s : finset ι) (f : ι → K) {j : ι} (hjs : j s) (hjf : f j 0) :
∃ (a : K), (∀ (i : ι), i sis_localization.is_integer A (a * f i)) ∃ (i : ι) (H : i s), a * f i J

Strengthening of is_localization.exist_integer_multiples: Let J ≠ ⊤ be an ideal in a Dedekind domain A, and f ≠ 0 a finite collection of elements of K = Frac(A), then we can multiply the elements of f by some a : K to find a collection of elements of A that is not completely contained in J.

theorem normalized_factors_prod {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {α : multiset (ideal T)} (h : ∀ (p : ideal T), p αprime p) :
theorem irreducible_pow_sup_of_le {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {I J : ideal T} (hJ : irreducible J) (n : ) (hn : n multiplicity J I) :
J ^ n I = J ^ n
theorem irreducible_pow_sup_of_ge {T : Type u_4} [comm_ring T] [is_domain T] [is_dedekind_domain T] {I J : ideal T} (hI : I ) (hJ : irreducible J) (n : ) (hn : multiplicity J I n) :
J ^ n I = J ^ (multiplicity J I).get _

Height one spectrum of a Dedekind domain #

If R is a Dedekind domain of Krull dimension 1, the maximal ideals of R are exactly its nonzero prime ideals. We define height_one_spectrum and provide lemmas to recover the facts that prime ideals of height one are prime and irreducible.

theorem is_dedekind_domain.height_one_spectrum.ext {R : Type u_1} {_inst_1 : comm_ring R} {_inst_5 : is_domain R} {_inst_6 : is_dedekind_domain R} (x y : is_dedekind_domain.height_one_spectrum R) (h : x.as_ideal = y.as_ideal) :
x = y
@[nolint, ext]
structure is_dedekind_domain.height_one_spectrum (R : Type u_1) [comm_ring R] [is_domain R] [is_dedekind_domain R] :
Type u_1

The height one prime spectrum of a Dedekind domain R is the type of nonzero prime ideals of R. Note that this equals the maximal spectrum if R has Krull dimension 1.

Instances for is_dedekind_domain.height_one_spectrum
  • is_dedekind_domain.height_one_spectrum.has_sizeof_inst
theorem is_dedekind_domain.height_one_spectrum.ext_iff {R : Type u_1} {_inst_1 : comm_ring R} {_inst_5 : is_domain R} {_inst_6 : is_dedekind_domain R} (x y : is_dedekind_domain.height_one_spectrum R) :
theorem ring.dimension_le_one.prime_le_prime_iff_eq {R : Type u_1} [comm_ring R] (h : ring.dimension_le_one R) {P Q : ideal R} [hP : P.is_prime] [hQ : Q.is_prime] (hP0 : P ) :
P Q P = Q
theorem ideal.coprime_of_no_prime_ge {R : Type u_1} [comm_ring R] {I J : ideal R} (h : ∀ (P : ideal R), I PJ P¬P.is_prime) :
I J =
theorem ideal.is_prime.mul_mem_pow {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] (I : ideal R) [hI : I.is_prime] {a b : R} {n : } (h : a * b I ^ n) :
a I b I ^ n
theorem ideal.count_normalized_factors_eq {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {p x : ideal R} [hp : p.is_prime] {n : } (hle : x p ^ n) (hlt : ¬x p ^ (n + 1)) :
theorem ideal.le_mul_of_no_prime_factors {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {I J K : ideal R} (coprime : ∀ (P : ideal R), J PK P¬P.is_prime) (hJ : I J) (hK : I K) :
I J * K
theorem ideal.le_of_pow_le_prime {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {I P : ideal R} [hP : P.is_prime] {n : } (h : I ^ n P) :
I P
theorem ideal.pow_le_prime_iff {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {I P : ideal R} [hP : P.is_prime] {n : } (hn : n 0) :
I ^ n P I P
theorem ideal.prod_le_prime {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {ι : Type u_2} {s : finset ι} {f : ι → ideal R} {P : ideal R} [hP : P.is_prime] :
s.prod (λ (i : ι), f i) P ∃ (i : ι) (H : i s), f i P
theorem is_dedekind_domain.inf_prime_pow_eq_prod {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {ι : Type u_2} (s : finset ι) (f : ι → ideal R) (e : ι → ) (prime : ∀ (i : ι), i s_root_.prime (f i)) (coprime : ∀ (i : ι), i s∀ (j : ι), j si jf i f j) :
s.inf (λ (i : ι), f i ^ e i) = s.prod (λ (i : ι), f i ^ e i)

The intersection of distinct prime powers in a Dedekind domain is the product of these prime powers.

noncomputable def is_dedekind_domain.quotient_equiv_pi_of_prod_eq {R : Type u_1} [comm_ring R] [is_domain R] [is_dedekind_domain R] {ι : Type u_2} [fintype ι] (I : ideal R) (P : ι → ideal R) (e : ι → ) (prime : ∀ (i : ι), _root_.prime (P i)) (coprime : ∀ (i j : ι), i jP i P j) (prod_eq : finset.univ.prod (λ (i : ι), P i ^ e i) = I) :
R I ≃+* Π (i : ι), R P i ^ e i

Chinese remainder theorem for a Dedekind domain: if the ideal I factors as ∏ i, P i ^ e i, then R ⧸ I factors as Π i, R ⧸ (P i ^ e i).

Equations

Chinese remainder theorem for a Dedekind domain: R ⧸ I factors as Π i, R ⧸ (P i ^ e i), where P i ranges over the prime factors of I and e i over the multiplicities.

Equations