statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
coe_ideal_span_singleton_mul_inv {x : R₁} (hx : x ≠ 0) :
(ideal.span ({x} : set R₁) : fractional_ideal R₁⁰ K) * (ideal.span ({x} : set R₁))⁻¹ = 1 | by rw [coe_ideal_span_singleton, span_singleton_mul_inv K $
(map_ne_zero_iff _ $ no_zero_smul_divisors.algebra_map_injective R₁ K).mpr hx] | lemma | fractional_ideal.coe_ideal_span_singleton_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"ideal.span",
"no_zero_smul_divisors.algebra_map_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_inv_mul {x : K} (hx : x ≠ 0) :
(span_singleton R₁⁰ x)⁻¹ * span_singleton R₁⁰ x = 1 | by rw [mul_comm, span_singleton_mul_inv K hx] | lemma | fractional_ideal.span_singleton_inv_mul | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_span_singleton_inv_mul {x : R₁} (hx : x ≠ 0) :
(ideal.span ({x} : set R₁) : fractional_ideal R₁⁰ K)⁻¹ * ideal.span ({x} : set R₁) = 1 | by rw [mul_comm, coe_ideal_span_singleton_mul_inv K hx] | lemma | fractional_ideal.coe_ideal_span_singleton_inv_mul | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"ideal.span",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_generator_self_inv {R₁ : Type*} [comm_ring R₁] [algebra R₁ K] [is_localization R₁⁰ K]
(I : fractional_ideal R₁⁰ K) [submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) :
I * span_singleton _ (generator (I : submodule R₁ K))⁻¹ = 1 | begin
-- Rewrite only the `I` that appears alone.
conv_lhs { congr, rw eq_span_singleton_of_principal I },
rw [span_singleton_mul_span_singleton, mul_inv_cancel, span_singleton_one],
intro generator_I_eq_zero,
apply h,
rw [eq_span_singleton_of_principal I, generator_I_eq_zero, span_singleton_zero]
end | lemma | fractional_ideal.mul_generator_self_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra",
"comm_ring",
"fractional_ideal",
"is_localization",
"mul_inv_cancel",
"submodule",
"submodule.is_principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_of_principal (I : fractional_ideal R₁⁰ K)
[submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) : I * I⁻¹ = 1 | (mul_div_self_cancel_iff).mpr
⟨span_singleton _ (generator (I : submodule R₁ K))⁻¹, mul_generator_self_inv _ I h⟩ | lemma | fractional_ideal.invertible_of_principal | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"submodule",
"submodule.is_principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_iff_generator_nonzero (I : fractional_ideal R₁⁰ K)
[submodule.is_principal (I : submodule R₁ K)] :
I * I⁻¹ = 1 ↔ generator (I : submodule R₁ K) ≠ 0 | begin
split,
{ intros hI hg,
apply ne_zero_of_mul_eq_one _ _ hI,
rw [eq_span_singleton_of_principal I, hg, span_singleton_zero] },
{ intro hg,
apply invertible_of_principal,
rw [eq_span_singleton_of_principal I],
intro hI,
have := mem_span_singleton_self _ (generator (I : submodule R₁ K)),... | lemma | fractional_ideal.invertible_iff_generator_nonzero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"submodule",
"submodule.is_principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_principal_inv (I : fractional_ideal R₁⁰ K)
[submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) :
submodule.is_principal (I⁻¹).1 | begin
rw [val_eq_coe, is_principal_iff],
use (generator (I : submodule R₁ K))⁻¹,
have hI : I * span_singleton _ ((generator (I : submodule R₁ K))⁻¹) = 1,
apply mul_generator_self_inv _ I h,
exact (right_inverse_eq _ I (span_singleton _ ((generator (I : submodule R₁ K))⁻¹)) hI).symm
end | lemma | fractional_ideal.is_principal_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"submodule",
"submodule.is_principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_dedekind_domain_inv : Prop | ∀ I ≠ (⊥ : fractional_ideal A⁰ (fraction_ring A)), I * I⁻¹ = 1 | def | is_dedekind_domain_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fraction_ring",
"fractional_ideal"
] | 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_dedekin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain_inv_iff [algebra A K] [is_fraction_ring A K] :
is_dedekind_domain_inv A ↔ (∀ I ≠ (⊥ : fractional_ideal A⁰ K), I * I⁻¹ = 1) | begin
let h := map_equiv (fraction_ring.alg_equiv A K),
refine h.to_equiv.forall_congr (λ I, _),
rw ← h.to_equiv.apply_eq_iff_eq,
simp [is_dedekind_domain_inv, show ⇑h.to_equiv = h, from rfl],
end | lemma | is_dedekind_domain_inv_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra",
"fraction_ring.alg_equiv",
"fractional_ideal",
"is_dedekind_domain_inv",
"is_fraction_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.adjoin_integral_eq_one_of_is_unit [algebra A K] [is_fraction_ring A K]
(x : K) (hx : is_integral A x) (hI : is_unit (adjoin_integral A⁰ x hx)) :
adjoin_integral A⁰ x hx = 1 | begin
set I := adjoin_integral A⁰ x hx,
have mul_self : I * I = I,
{ apply coe_to_submodule_injective, simp },
convert congr_arg (* I⁻¹) mul_self;
simp only [(mul_inv_cancel_iff_is_unit K).mpr hI, mul_assoc, mul_one],
end | lemma | fractional_ideal.adjoin_integral_eq_one_of_is_unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra",
"is_fraction_ring",
"is_integral",
"is_unit",
"mul_assoc",
"mul_one",
"mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_one {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : I * I⁻¹ = 1 | is_dedekind_domain_inv_iff.mp h I hI | lemma | is_dedekind_domain_inv.mul_inv_eq_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"mul_inv_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_one {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : I⁻¹ * I = 1 | (mul_comm _ _).trans (h.mul_inv_eq_one hI) | lemma | is_dedekind_domain_inv.inv_mul_eq_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"inv_mul_eq_one",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : is_unit I | is_unit_of_mul_eq_one _ _ (h.mul_inv_eq_one hI) | lemma | is_dedekind_domain_inv.is_unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_unit",
"is_unit_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring : is_noetherian_ring A | begin
refine is_noetherian_ring_iff.mpr ⟨λ (I : ideal A), _⟩,
by_cases hI : I = ⊥,
{ rw hI, apply submodule.fg_bot },
have hI : (I : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 := coe_ideal_ne_zero.mpr hI,
exact I.fg_of_is_unit (is_fraction_ring.injective A (fraction_ring A)) (h.is_unit hI)
end | lemma | is_dedekind_domain_inv.is_noetherian_ring | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fraction_ring",
"fractional_ideal",
"ideal",
"is_fraction_ring.injective",
"is_noetherian_ring",
"submodule.fg_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrally_closed : is_integrally_closed A | begin
-- It suffices to show that for integral `x`,
-- `A[x]` (which is a fractional ideal) is in fact equal to `A`.
refine ⟨λ x hx, _⟩,
rw [← set.mem_range, ← algebra.mem_bot, ← subalgebra.mem_to_submodule, algebra.to_submodule_bot,
← coe_span_singleton A⁰ (1 : fraction_ring A), span_singleton_one,
... | lemma | is_dedekind_domain_inv.integrally_closed | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra.mem_bot",
"algebra.to_submodule_bot",
"fraction_ring",
"fractional_ideal.adjoin_integral_eq_one_of_is_unit",
"is_integrally_closed",
"one_ne_zero",
"set.mem_range",
"subalgebra.mem_to_submodule",
"subalgebra.one_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimension_le_one : dimension_le_one A | begin
-- We're going to show that `P` is maximal because any (maximal) ideal `M`
-- that is strictly larger would be `⊤`.
rintros P P_ne hP,
refine ideal.is_maximal_def.mpr ⟨hP.ne_top, λ M hM, _⟩,
-- We may assume `P` and `M` (as fractional ideals) are nonzero.
have P'_ne : (P : fractional_ideal A⁰ (fractio... | lemma | is_dedekind_domain_inv.dimension_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"bot_le",
"eq_top_iff",
"fraction_ring",
"fractional_ideal",
"is_fraction_ring.injective",
"mul_assoc",
"one_mul",
"ring_hom.map_mul",
"set_like.exists_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_dedekind_domain : is_dedekind_domain A | ⟨h.is_noetherian_ring, h.dimension_le_one, h.integrally_closed⟩ | theorem | is_dedekind_domain_inv.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"is_dedekind_domain"
] | Showing one side of the equivalence between the definitions
`is_dedekind_domain_inv` and `is_dedekind_domain` of Dedekind domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_multiset_prod_cons_le_and_prod_not_le [is_dedekind_domain A]
(hNF : ¬ is_field A) {I M : ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.is_maximal] :
∃ (Z : multiset (prime_spectrum A)),
(M ::ₘ (Z.map prime_spectrum.as_ideal)).prod ≤ I ∧
¬ (multiset.prod (Z.map prime_spectrum.as_ideal) ≤ I) | begin
-- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
obtain ⟨Z₀, hZ₀⟩ := prime_spectrum.exists_prime_spectrum_prod_le_and_ne_bot_of_domain hNF hI0,
obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ := multiset.well_founded_lt.has_min
(λ Z, (Z.map prime_spectrum.as_ideal).prod ≤ I ∧... | lemma | exists_multiset_prod_cons_le_and_prod_not_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.mul_eq_bot",
"is_dedekind_domain",
"is_field",
"multiset",
"multiset.cons_erase",
"multiset.map_erase",
"multiset.prod",
"multiset.prod_cons",
"not_or_distrib",
"prime_spectrum",
"prime_spectrum.exists_prime_spectrum_prod_le_and_ne_bot_of_domain"
] | 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 ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_not_mem_one_of_ne_bot [is_dedekind_domain A]
(hNF : ¬ is_field A) {I : ideal A} (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
∃ x : K, x ∈ (I⁻¹ : fractional_ideal A⁰ K) ∧ x ∉ (1 : fractional_ideal A⁰ K) | begin
-- WLOG, let `I` be maximal.
suffices : ∀ {M : ideal A} (hM : M.is_maximal),
∃ x : K, x ∈ (M⁻¹ : fractional_ideal A⁰ K) ∧ x ∉ (1 : fractional_ideal A⁰ K),
{ obtain ⟨M, hM, hIM⟩ : ∃ (M : ideal A), is_maximal M ∧ I ≤ M := ideal.exists_le_maximal I hI1,
resetI,
have hM0 := (M.bot_lt_of_maximal hNF)... | lemma | fractional_ideal.exists_not_mem_one_of_ne_bot | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra_map",
"div_eq_mul_inv",
"eq_div_iff_mul_eq",
"exists_multiset_prod_cons_le_and_prod_not_le",
"fractional_ideal",
"ideal",
"ideal.exists_le_maximal",
"ideal.mem_span_singleton'",
"ideal.span",
"is_dedekind_domain",
"is_field",
"is_fraction_ring.injective",
"mul_assoc",
"mul_comm",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem_inv_coe_ideal {I : ideal A} (hI : I ≠ ⊥) :
(1 : K) ∈ (I : fractional_ideal A⁰ K)⁻¹ | begin
rw mem_inv_iff (coe_ideal_ne_zero.mpr hI),
intros y hy,
rw one_mul,
exact coe_ideal_le_one hy,
assumption
end | lemma | fractional_ideal.one_mem_inv_coe_ideal | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"ideal",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_of_le_one [h : is_dedekind_domain A]
{I : ideal A} (hI0 : I ≠ ⊥) (hI : ((I * I⁻¹)⁻¹ : fractional_ideal A⁰ K) ≤ 1) :
(I * I⁻¹ : fractional_ideal A⁰ K) = 1 | begin
-- Handle a few trivial cases.
by_cases hI1 : I = ⊤,
{ rw [hI1, coe_ideal_top, one_mul, inv_one] },
by_cases hNF : is_field A,
{ letI := hNF.to_field, rcases hI1 (I.eq_bot_or_top.resolve_left hI0) },
-- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
-- `J⁻¹ = (I * I⁻¹)⁻¹` cannot hav... | lemma | fractional_ideal.mul_inv_cancel_of_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"eq_bot_iff",
"fractional_ideal",
"ideal",
"inv_one",
"is_dedekind_domain",
"is_field",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_mul_inv [h : is_dedekind_domain A] (I : ideal A) (hI0 : I ≠ ⊥) :
(I * I⁻¹ : fractional_ideal A⁰ K) = 1 | begin
-- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
apply mul_inv_cancel_of_le_one hI0,
by_cases hJ0 : (I * I⁻¹ : fractional_ideal A⁰ K) = 0,
{ rw [hJ0, inv_zero'], exact zero_le _ },
intros x hx,
-- In particular, we'll show all `x ∈ J⁻¹` are integral.
suffices : x ∈ integral_closure A K,
{ rwa [is_integr... | lemma | fractional_ideal.coe_ideal_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"alg_hom.mem_range",
"algebra.mem_bot",
"fractional_ideal",
"ideal",
"ih",
"integral_closure",
"is_dedekind_domain",
"is_integrally_closed.integral_closure_eq_bot",
"is_noetherian",
"mem_integral_closure_iff_mem_fg",
"mul_assoc",
"mul_comm",
"polynomial.aeval",
"polynomial.aeval_X",
"pol... | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel [is_dedekind_domain A]
{I : fractional_ideal A⁰ K} (hne : I ≠ 0) : I * I⁻¹ = 1 | begin
obtain ⟨a, J, ha, hJ⟩ :
∃ (a : A) (aI : ideal A), a ≠ 0 ∧ I = span_singleton A⁰ (algebra_map _ _ a)⁻¹ * aI :=
exists_eq_span_singleton_mul I,
suffices h₂ : I * (span_singleton A⁰ (algebra_map _ _ a) * J⁻¹) = 1,
{ rw mul_inv_cancel_iff,
exact ⟨span_singleton A⁰ (algebra_map _ _ a) * J⁻¹, h₂⟩ },
... | theorem | fractional_ideal.mul_inv_cancel | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra_map",
"fractional_ideal",
"ideal",
"inv_mul_cancel",
"is_dedekind_domain",
"is_fraction_ring.injective",
"mul_assoc",
"mul_inv_cancel",
"mul_left_comm",
"mul_one",
"right_ne_zero_of_mul"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_le_iff [is_dedekind_domain A] {J : fractional_ideal A⁰ K}
(hJ : J ≠ 0) : ∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' | begin
intros I I',
split,
{ intros h, convert mul_right_mono J⁻¹ h;
rw [mul_assoc, fractional_ideal.mul_inv_cancel hJ, mul_one] },
{ exact λ h, mul_right_mono J h }
end | lemma | fractional_ideal.mul_right_le_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"fractional_ideal.mul_inv_cancel",
"is_dedekind_domain",
"mul_assoc",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_le_iff [is_dedekind_domain A] {J : fractional_ideal A⁰ K}
(hJ : J ≠ 0) {I I'} : J * I ≤ J * I' ↔ I ≤ I' | by convert mul_right_le_iff hJ using 1; simp only [mul_comm] | lemma | fractional_ideal.mul_left_le_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_dedekind_domain",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_strict_mono [is_dedekind_domain A] {I : fractional_ideal A⁰ K}
(hI : I ≠ 0) : strict_mono (* I) | strict_mono_of_le_iff_le (λ _ _, (mul_right_le_iff hI).symm) | lemma | fractional_ideal.mul_right_strict_mono | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_dedekind_domain",
"strict_mono",
"strict_mono_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_strict_mono [is_dedekind_domain A] {I : fractional_ideal A⁰ K}
(hI : I ≠ 0) : strict_mono ((*) I) | strict_mono_of_le_iff_le (λ _ _, (mul_left_le_iff hI).symm) | lemma | fractional_ideal.mul_left_strict_mono | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_dedekind_domain",
"strict_mono",
"strict_mono_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_mul_inv [is_dedekind_domain A] (I J : fractional_ideal A⁰ K) :
I / J = I * J⁻¹ | begin
by_cases hJ : J = 0,
{ rw [hJ, div_zero, inv_zero', mul_zero] },
refine le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _),
{ rw [mul_assoc, mul_comm J⁻¹, fractional_ideal.mul_inv_cancel hJ, mul_one, mul_le],
intros x hx y hy,
rw [mem_div_iff_of_nonzero hJ] at hx,
exact hx... | lemma | fractional_ideal.div_eq_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_eq_mul_inv",
"div_zero",
"fractional_ideal",
"fractional_ideal.mul_inv_cancel",
"is_dedekind_domain",
"le_div_iff_mul_le",
"mul_assoc",
"mul_comm",
"mul_one",
"mul_zero"
] | This is also available as `_root_.div_eq_mul_inv`, using the
`comm_group_with_zero` instance defined below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain_iff_is_dedekind_domain_inv :
is_dedekind_domain A ↔ is_dedekind_domain_inv A | ⟨λ h I hI, by exactI fractional_ideal.mul_inv_cancel hI, λ h, h.is_dedekind_domain⟩ | theorem | is_dedekind_domain_iff_is_dedekind_domain_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal.mul_inv_cancel",
"is_dedekind_domain",
"is_dedekind_domain_inv"
] | `is_dedekind_domain` and `is_dedekind_domain_inv` are equivalent ways
to express that an integral domain is a Dedekind domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fractional_ideal.semifield :
semifield (fractional_ideal A⁰ K) | { inv := λ I, I⁻¹,
inv_zero := inv_zero' _,
div := (/),
div_eq_mul_inv := fractional_ideal.div_eq_mul_inv,
mul_inv_cancel := λ I, fractional_ideal.mul_inv_cancel,
.. fractional_ideal.comm_semiring, .. coe_ideal_injective.nontrivial } | instance | fractional_ideal.semifield | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_eq_mul_inv",
"fractional_ideal",
"fractional_ideal.div_eq_mul_inv",
"fractional_ideal.mul_inv_cancel",
"inv_zero",
"mul_inv_cancel",
"semifield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.cancel_comm_monoid_with_zero :
cancel_comm_monoid_with_zero (fractional_ideal A⁰ K) | { .. fractional_ideal.comm_semiring, -- Project out the computable fields first.
.. (by apply_instance : cancel_comm_monoid_with_zero (fractional_ideal A⁰ K)) } | instance | fractional_ideal.cancel_comm_monoid_with_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"cancel_comm_monoid_with_zero",
"fractional_ideal"
] | Fractional ideals have cancellative multiplication in a Dedekind domain.
Although this instance is a direct consequence of the instance
`fractional_ideal.comm_group_with_zero`, we define this instance to provide
a computable alternative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.cancel_comm_monoid_with_zero :
cancel_comm_monoid_with_zero (ideal A) | { .. ideal.idem_comm_semiring,
.. function.injective.cancel_comm_monoid_with_zero (coe_ideal_hom A⁰ (fraction_ring A))
coe_ideal_injective (ring_hom.map_zero _) (ring_hom.map_one _) (ring_hom.map_mul _)
(ring_hom.map_pow _) } | instance | ideal.cancel_comm_monoid_with_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"cancel_comm_monoid_with_zero",
"fraction_ring",
"function.injective.cancel_comm_monoid_with_zero",
"ideal",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_pow",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_domain :
is_domain (ideal A) | { .. (infer_instance : is_cancel_mul_zero _), .. ideal.nontrivial } | instance | ideal.is_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"is_cancel_mul_zero",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.dvd_iff_le {I J : ideal A} : (I ∣ J) ↔ J ≤ I | ⟨ideal.le_of_dvd,
λ h, begin
by_cases hI : I = ⊥,
{ have hJ : J = ⊥, { rwa [hI, ← eq_bot_iff] at h },
rw [hI, hJ] },
have hI' : (I : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 := coe_ideal_ne_zero.mpr hI,
have : (I : fractional_ideal A⁰ (fraction_ring A))⁻¹ * J ≤ 1 := le_trans
(mul_left_mo... | lemma | ideal.dvd_iff_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"eq_bot_iff",
"fraction_ring",
"fractional_ideal",
"ideal",
"inv_mul_cancel",
"mul_assoc",
"mul_inv_cancel",
"one_mul"
] | For ideals in a Dedekind domain, to divide is to contain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.dvd_not_unit_iff_lt {I J : ideal A} :
dvd_not_unit I J ↔ J < I | ⟨λ ⟨hI, H, hunit, hmul⟩, lt_of_le_of_ne (ideal.dvd_iff_le.mp ⟨H, hmul⟩)
(mt (λ h, have H = 1, from mul_left_cancel₀ hI (by rw [← hmul, h, mul_one]),
show is_unit H, from this.symm ▸ is_unit_one) hunit),
λ h, dvd_not_unit_of_dvd_of_not_dvd (ideal.dvd_iff_le.mpr (le_of_lt h))
(mt ideal.dvd_iff_le.mp (not_le_of_... | lemma | ideal.dvd_not_unit_iff_lt | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"dvd_not_unit",
"dvd_not_unit_of_dvd_of_not_dvd",
"ideal",
"is_unit",
"is_unit_one",
"mul_left_cancel₀",
"mul_one",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.unique_factorization_monoid :
unique_factorization_monoid (ideal A) | { irreducible_iff_prime := λ P,
⟨λ hirr, ⟨hirr.ne_zero, hirr.not_unit, λ I J, begin
have : P.is_maximal,
{ refine ⟨⟨mt ideal.is_unit_iff.mpr hirr.not_unit, _⟩⟩,
intros J hJ,
obtain ⟨J_ne, H, hunit, P_eq⟩ := ideal.dvd_not_unit_iff_lt.mpr hJ,
exact ideal.is_unit_iff.mp ((hirr.is_unit_or_is_uni... | instance | ideal.unique_factorization_monoid | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.mul_mem_mul",
"set_like.le_def",
"unique_factorization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.normalization_monoid : normalization_monoid (ideal A) | normalization_monoid_of_unique_units | instance | ideal.normalization_monoid | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"normalization_monoid",
"normalization_monoid_of_unique_units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.dvd_span_singleton {I : ideal A} {x : A} :
I ∣ ideal.span {x} ↔ x ∈ I | ideal.dvd_iff_le.trans (ideal.span_le.trans set.singleton_subset_iff) | lemma | ideal.dvd_span_singleton | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.span",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_prime_of_prime {P : ideal A} (h : prime P) : is_prime P | begin
refine ⟨_, λ x y hxy, _⟩,
{ unfreezingI { rintro rfl },
rw ← ideal.one_eq_top at h,
exact h.not_unit is_unit_one },
{ simp only [← ideal.dvd_span_singleton, ← ideal.span_singleton_mul_span_singleton] at ⊢ hxy,
exact h.dvd_or_dvd hxy }
end | lemma | ideal.is_prime_of_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_span_singleton",
"ideal.one_eq_top",
"ideal.span_singleton_mul_span_singleton",
"is_unit_one",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.prime_of_is_prime {P : ideal A} (hP : P ≠ ⊥) (h : is_prime P) : prime P | begin
refine ⟨hP, mt ideal.is_unit_iff.mp h.ne_top, λ I J hIJ, _⟩,
simpa only [ideal.dvd_iff_le] using (h.mul_le.mp (ideal.le_of_dvd hIJ)),
end | theorem | ideal.prime_of_is_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.le_of_dvd",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.prime_iff_is_prime {P : ideal A} (hP : P ≠ ⊥) :
prime P ↔ is_prime P | ⟨ideal.is_prime_of_prime, ideal.prime_of_is_prime hP⟩ | theorem | ideal.prime_iff_is_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.prime_of_is_prime",
"prime"
] | In a Dedekind domain, the (nonzero) prime elements of the monoid with zero `ideal A`
are exactly the prime ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.is_prime_iff_bot_or_prime {P : ideal A} :
is_prime P ↔ P = ⊥ ∨ prime P | ⟨λ hp, (eq_or_ne P ⊥).imp_right $ λ hp0, (ideal.prime_of_is_prime hp0 hp),
λ hp, hp.elim (λ h, h.symm ▸ ideal.bot_prime) ideal.is_prime_of_prime⟩ | theorem | ideal.is_prime_iff_bot_or_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"eq_or_ne",
"ideal",
"ideal.bot_prime",
"ideal.prime_of_is_prime",
"prime"
] | In a Dedekind domain, the the prime ideals are the zero ideal together with the prime elements
of the monoid with zero `ideal A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.strict_anti_pow (I : ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
strict_anti ((^) I : ℕ → ideal A) | strict_anti_nat_of_succ_lt $ λ e, ideal.dvd_not_unit_iff_lt.mp
⟨pow_ne_zero _ hI0, I, mt is_unit_iff.mp hI1, pow_succ' I e⟩ | lemma | ideal.strict_anti_pow | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"pow_succ'",
"strict_anti",
"strict_anti_nat_of_succ_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.pow_lt_self (I : ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) : I^e < I | by convert I.strict_anti_pow hI0 hI1 he; rw pow_one | lemma | ideal.pow_lt_self | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exists_mem_pow_not_mem_pow_succ (I : ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) :
∃ x ∈ I^e, x ∉ I^(e+1) | set_like.exists_of_lt (I.strict_anti_pow hI0 hI1 e.lt_succ_self) | lemma | ideal.exists_mem_pow_not_mem_pow_succ | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"set_like.exists_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.eq_prime_pow_of_succ_lt_of_le {P I : ideal A} [P_prime : P.is_prime] (hP : P ≠ ⊥)
{i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) :
I = P ^ i | begin
letI := classical.dec_eq (ideal A),
refine le_antisymm hle _,
have P_prime' := ideal.prime_of_is_prime hP P_prime,
have : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne',
have := pow_ne_zero i hP,
have := pow_ne_zero (i + 1) hP,
rw [← ideal.dvd_not_unit_iff_lt, dvd_not_unit_iff_normalized_factors_lt_normal... | lemma | ideal.eq_prime_pow_of_succ_lt_of_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"bot_le",
"classical.dec_eq",
"ideal",
"ideal.dvd_iff_le",
"ideal.dvd_not_unit_iff_lt",
"ideal.prime_of_is_prime",
"multiset.lt_replicate_succ",
"multiset.nsmul_singleton",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.pow_succ_lt_pow {P : ideal A} [P_prime : P.is_prime] (hP : P ≠ ⊥)
(i : ℕ) :
P ^ (i + 1) < P ^ i | lt_of_le_of_ne (ideal.pow_le_pow (nat.le_succ _))
(mt (pow_eq_pow_iff hP (mt ideal.is_unit_iff.mp P_prime.ne_top)).mp i.succ_ne_self) | lemma | ideal.pow_succ_lt_pow | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.pow_le_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates.le_singleton_iff (x : A) (n : ℕ) (I : ideal A) :
associates.mk I^n ≤ associates.mk (ideal.span {x}) ↔ x ∈ I^n | begin
rw [← associates.dvd_eq_le, ← associates.mk_pow, associates.mk_dvd_mk, ideal.dvd_span_singleton],
end | lemma | associates.le_singleton_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"associates.dvd_eq_le",
"associates.mk",
"associates.mk_dvd_mk",
"associates.mk_pow",
"ideal",
"ideal.dvd_span_singleton",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exist_integer_multiples_not_mem
{J : ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : finset ι) (f : ι → K)
{j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
∃ a : K, (∀ i ∈ s, is_localization.is_integer A (a * f i)) ∧
∃ i ∈ s, (a * f i) ∉ (J : fractional_ideal A⁰ K) | begin
-- Consider the fractional ideal `I` spanned by the `f`s.
let I : fractional_ideal A⁰ K := span_finset A s f,
have hI0 : I ≠ 0 := span_finset_ne_zero.mpr ⟨j, hjs, hjf⟩,
-- We claim the multiplier `a` we're looking for is in `I⁻¹ \ (J / I)`.
suffices : ↑J / I < I⁻¹,
{ obtain ⟨_, a, hI, hpI⟩ := set_like... | lemma | ideal.exist_integer_multiples_not_mem | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_eq_mul_inv",
"finset",
"fractional_ideal",
"ideal",
"inv_ne_zero",
"is_localization.is_integer",
"mul_smul_comm",
"mul_zero",
"one_mul",
"set.mem_image_of_mem",
"strict_mono_of_le_iff_le",
"submodule.add_mem",
"submodule.smul_mem",
"submodule.span_induction",
"submodule.subset_span"... | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_mul_inf (I J : ideal A) : (I ⊔ J) * (I ⊓ J) = I * J | begin
letI := classical.dec_eq (ideal A),
letI := classical.dec_eq (associates (ideal A)),
letI := unique_factorization_monoid.to_normalized_gcd_monoid (ideal A),
have hgcd : gcd I J = I ⊔ J,
{ rw [gcd_eq_normalize _ _, normalize_eq],
{ rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le],
exact ... | lemma | ideal.sup_mul_inf | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"associates",
"classical.dec_eq",
"dvd_gcd_iff",
"dvd_lcm_right",
"gcd_eq_normalize",
"gcd_mul_lcm",
"ideal",
"lcm_dvd_iff",
"lcm_eq_normalize",
"le_inf_iff",
"normalize_eq",
"sup_le_iff",
"unique_factorization_monoid.to_normalized_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_sup (I J : ideal A) : gcd I J = I ⊔ J | rfl | lemma | ideal.gcd_eq_sup | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_inf (I J : ideal A) : lcm I J = I ⊓ J | rfl | lemma | ideal.lcm_eq_inf | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq_mul_of_coprime {I J : ideal A} (coprime : I ⊔ J = ⊤) :
I ⊓ J = I * J | by rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime, top_mul] | lemma | ideal.inf_eq_mul_of_coprime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"gcd_mul_lcm",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_normalized_factors_eq_self (hI : I ≠ ⊥) : (normalized_factors I).prod = I | associated_iff_eq.1 (normalized_factors_prod hI) | lemma | prod_normalized_factors_eq_self | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
count_le_of_ideal_ge {I J : ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K : ideal T) :
count K (normalized_factors J) ≤ count K (normalized_factors I) | le_iff_count.1 ((dvd_iff_normalized_factors_le_normalized_factors (ne_bot_of_le_ne_bot hI h) hI).1
(dvd_iff_le.2 h)) _ | lemma | count_le_of_ideal_ge | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ne_bot_of_le_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
I ⊔ J = (normalized_factors I ∩ normalized_factors J).prod | begin
have H : normalized_factors (normalized_factors I ∩ normalized_factors J).prod =
normalized_factors I ∩ normalized_factors J,
{ apply normalized_factors_prod_of_prime,
intros p hp,
rw mem_inter at hp,
exact prime_of_normalized_factor p hp.left },
have := (multiset.prod_ne_zero_of_prime (norm... | lemma | sup_eq_prod_inf_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"count_le_of_ideal_ge",
"inf_le_left",
"inf_le_right",
"le_sup_left",
"le_sup_right",
"multiset.count_inter",
"multiset.prod_ne_zero_of_prime",
"ne_bot_of_le_ne_bot",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_pow_sup (hI : I ≠ ⊥) (hJ : irreducible J) (n : ℕ) :
J^n ⊔ I = J^(min ((normalized_factors I).count J) n) | by rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
normalized_factors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate] | lemma | irreducible_pow_sup | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"irreducible",
"normalize_eq",
"pow_ne_zero",
"sup_eq_prod_inf_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_pow_sup_of_le (hJ : irreducible J) (n : ℕ)
(hn : ↑n ≤ multiplicity J I) : J^n ⊔ I = J^n | begin
by_cases hI : I = ⊥,
{ simp [*] at *, },
rw [irreducible_pow_sup hI hJ, min_eq_right],
rwa [multiplicity_eq_count_normalized_factors hJ hI, part_enat.coe_le_coe, normalize_eq J] at hn
end | lemma | irreducible_pow_sup_of_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"irreducible",
"irreducible_pow_sup",
"multiplicity",
"normalize_eq",
"part_enat.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_pow_sup_of_ge (hI : I ≠ ⊥) (hJ : irreducible J) (n : ℕ)
(hn : multiplicity J I ≤ n) : J^n ⊔ I = J ^ (multiplicity J I).get (part_enat.dom_of_le_coe hn) | begin
rw [irreducible_pow_sup hI hJ, min_eq_left],
congr,
{ rw [← part_enat.coe_inj, part_enat.coe_get, multiplicity_eq_count_normalized_factors hJ hI,
normalize_eq J] },
{ rwa [multiplicity_eq_count_normalized_factors hJ hI, part_enat.coe_le_coe, normalize_eq J]
at hn }
end | lemma | irreducible_pow_sup_of_ge | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"irreducible",
"irreducible_pow_sup",
"multiplicity",
"normalize_eq",
"part_enat.coe_get",
"part_enat.coe_inj",
"part_enat.coe_le_coe",
"part_enat.dom_of_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
height_one_spectrum | (as_ideal : ideal R)
(is_prime : as_ideal.is_prime)
(ne_bot : as_ideal ≠ ⊥) | structure | is_dedekind_domain.height_one_spectrum | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal : v.as_ideal.is_maximal | dimension_le_one v.as_ideal v.ne_bot v.is_prime | instance | is_dedekind_domain.height_one_spectrum.is_maximal | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime : prime v.as_ideal | ideal.prime_of_is_prime v.ne_bot v.is_prime | lemma | is_dedekind_domain.height_one_spectrum.prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal.prime_of_is_prime",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible : irreducible v.as_ideal | unique_factorization_monoid.irreducible_iff_prime.mpr v.prime | lemma | is_dedekind_domain.height_one_spectrum.irreducible | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates_irreducible : _root_.irreducible $ associates.mk v.as_ideal | (associates.irreducible_mk _).mpr v.irreducible | lemma | is_dedekind_domain.height_one_spectrum.associates_irreducible | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"associates.irreducible_mk",
"associates.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_maximal_spectrum (hR : ¬is_field R) : height_one_spectrum R ≃ maximal_spectrum R | { to_fun := λ v, ⟨v.as_ideal, dimension_le_one v.as_ideal v.ne_bot v.is_prime⟩,
inv_fun := λ v,
⟨v.as_ideal, v.is_maximal.is_prime, ring.ne_bot_of_is_maximal_of_not_is_field v.is_maximal hR⟩,
left_inv := λ ⟨_, _, _⟩, rfl,
right_inv := λ ⟨_, _⟩, rfl } | def | is_dedekind_domain.height_one_spectrum.equiv_maximal_spectrum | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"inv_fun",
"is_field",
"maximal_spectrum",
"ring.ne_bot_of_is_maximal_of_not_is_field"
] | An equivalence between the height one and maximal spectra for rings of Krull dimension 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_localization_eq_bot [algebra R K] [hK : is_fraction_ring R K] :
(⨅ v : height_one_spectrum R,
localization.subalgebra.of_field K _ v.as_ideal.prime_compl_le_non_zero_divisors) = ⊥ | begin
ext x,
rw [algebra.mem_infi],
split,
by_cases hR : is_field R,
{ rcases function.bijective_iff_has_inverse.mp
(is_field.localization_map_bijective (flip non_zero_divisors.ne_zero rfl : 0 ∉ R⁰) hR)
with ⟨algebra_map_inv, _, algebra_map_right_inv⟩,
exact λ _, algebra.mem_bot.mpr ⟨algebra_m... | theorem | is_dedekind_domain.height_one_spectrum.infi_localization_eq_bot | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"algebra",
"algebra.mem_infi",
"is_field",
"is_field.localization_map_bijective",
"is_fraction_ring",
"localization.subalgebra.of_field",
"maximal_spectrum.infi_localization_eq_bot",
"non_zero_divisors.ne_zero"
] | A Dedekind domain is equal to the intersection of its localizations at all its height one
non-zero prime ideals viewed as subalgebras of its field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_factors_fun_of_quot_hom {f : R ⧸ I →+* A ⧸ J} (hf : function.surjective f ) :
{p : ideal R | p ∣ I} →o {p : ideal A | p ∣ J} | { to_fun := λ X, ⟨comap J^.quotient.mk (map f (map I^.quotient.mk X)),
begin
have : (J^.quotient.mk).ker ≤ comap J^.quotient.mk (map f (map I^.quotient.mk X)),
{ exact ker_le_comap J^.quotient.mk },
rw mk_ker at this,
exact dvd_iff_le.mpr this,
end ⟩,
monotone' :=
begin
rintr... | def | ideal_factors_fun_of_quot_hom | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"le_sup_of_le_left",
"ring_hom.ker_eq_comap_bot",
"subtype.coe_le_coe",
"subtype.coe_mk"
] | The map from ideals of `R` dividing `I` to the ideals of `A` dividing `J` induced by
a homomorphism `f : R/I →+* A/J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_factors_fun_of_quot_hom_id :
ideal_factors_fun_of_quot_hom (ring_hom.id (A ⧸ J)).is_surjective = order_hom.id | order_hom.ext _ _ (funext $ λ X, by simp only [ideal_factors_fun_of_quot_hom, map_id,
order_hom.coe_fun_mk, order_hom.id_coe, id.def, comap_map_of_surjective J^.quotient.mk
quotient.mk_surjective, ← ring_hom.ker_eq_comap_bot J^.quotient.mk, mk_ker, sup_eq_left.mpr
(dvd_iff_le.mp X.prop), subtype.coe_eta] ) | lemma | ideal_factors_fun_of_quot_hom_id | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal_factors_fun_of_quot_hom",
"map_id",
"order_hom.coe_fun_mk",
"order_hom.ext",
"order_hom.id",
"ring_hom.id",
"ring_hom.ker_eq_comap_bot",
"subtype.coe_eta"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_factors_fun_of_quot_hom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
(hf : function.surjective f) (hg : function.surjective g) :
(ideal_factors_fun_of_quot_hom hg).comp (ideal_factors_fun_of_quot_hom hf)
= ideal_factors_fun_of_quot_hom (show function.surjective (g.comp f), from hg.comp hf) | begin
refine order_hom.ext _ _ (funext $ λ x, _),
rw [ideal_factors_fun_of_quot_hom, ideal_factors_fun_of_quot_hom, order_hom.comp_coe,
order_hom.coe_fun_mk, order_hom.coe_fun_mk, function.comp_app,
ideal_factors_fun_of_quot_hom, order_hom.coe_fun_mk, subtype.mk_eq_mk, subtype.coe_mk,
map_comap_of_surj... | lemma | ideal_factors_fun_of_quot_hom_comp | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal_factors_fun_of_quot_hom",
"order_hom.coe_fun_mk",
"order_hom.ext",
"subtype.coe_mk",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_factors_equiv_of_quot_equiv : {p : ideal R | p ∣ I} ≃o {p : ideal A | p ∣ J} | order_iso.of_hom_inv
(ideal_factors_fun_of_quot_hom (show function.surjective
(f : R ⧸I →+* A ⧸ J), from f.surjective))
(ideal_factors_fun_of_quot_hom (show function.surjective
(f.symm : A ⧸J →+* R ⧸ I), from f.symm.surjective))
(by simp only [← ideal_factors_fun_of_quot_hom_id, order_hom.coe_eq, order_... | def | ideal_factors_equiv_of_quot_equiv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal_factors_fun_of_quot_hom",
"ideal_factors_fun_of_quot_hom_comp",
"ideal_factors_fun_of_quot_hom_id",
"order_hom.coe_eq",
"order_iso.of_hom_inv",
"ring_equiv.self_trans_symm",
"ring_equiv.symm_trans_self",
"ring_equiv.to_ring_hom_eq_coe",
"ring_equiv.to_ring_hom_refl",
"ring_equiv.... | The bijection between ideals of `R` dividing `I` and the ideals of `A` dividing `J` induced by
an isomorphism `f : R/I ≅ A/J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_factors_equiv_of_quot_equiv_symm :
(ideal_factors_equiv_of_quot_equiv f).symm = ideal_factors_equiv_of_quot_equiv f.symm | rfl | lemma | ideal_factors_equiv_of_quot_equiv_symm | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal_factors_equiv_of_quot_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_factors_equiv_of_quot_equiv_is_dvd_iso {L M : ideal R} (hL : L ∣ I) (hM : M ∣ I) :
(ideal_factors_equiv_of_quot_equiv f ⟨L, hL⟩ : ideal A) ∣
ideal_factors_equiv_of_quot_equiv f ⟨M, hM⟩ ↔ L ∣ M | begin
suffices : ideal_factors_equiv_of_quot_equiv f ⟨M, hM⟩ ≤
ideal_factors_equiv_of_quot_equiv f ⟨L, hL⟩ ↔ (⟨M, hM⟩ : {p : ideal R | p ∣ I}) ≤ ⟨L, hL⟩,
{ rw [dvd_iff_le, dvd_iff_le, subtype.coe_le_coe, this, subtype.mk_le_mk] },
exact (ideal_factors_equiv_of_quot_equiv f).le_iff_le,
end | lemma | ideal_factors_equiv_of_quot_equiv_is_dvd_iso | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal_factors_equiv_of_quot_equiv",
"subtype.coe_le_coe",
"subtype.mk_le_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors
(hJ : J ≠ ⊥) {L : ideal R} (hL : L ∈ normalized_factors I) :
↑(ideal_factors_equiv_of_quot_equiv f
⟨L, dvd_of_mem_normalized_factors hL⟩) ∈ normalized_factors J | begin
by_cases hI : I = ⊥,
{ exfalso,
rw [hI, bot_eq_zero, normalized_factors_zero, ← multiset.empty_eq_zero] at hL,
exact hL, },
{ apply mem_normalized_factors_factor_dvd_iso_of_mem_normalized_factors hI hJ hL _,
rintros ⟨l, hl⟩ ⟨l', hl'⟩,
rw [subtype.coe_mk, subtype.coe_mk],
apply ideal_fact... | lemma | ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal_factors_equiv_of_quot_equiv",
"ideal_factors_equiv_of_quot_equiv_is_dvd_iso",
"mem_normalized_factors_factor_dvd_iso_of_mem_normalized_factors",
"multiset.empty_eq_zero",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalized_factors_equiv_of_quot_equiv (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
{L : ideal R | L ∈ normalized_factors I } ≃ {M : ideal A | M ∈ normalized_factors J } | { to_fun := λ j, ⟨ideal_factors_equiv_of_quot_equiv f ⟨↑j, dvd_of_mem_normalized_factors j.prop⟩,
ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors f hJ j.prop⟩,
inv_fun := λ j, ⟨(ideal_factors_equiv_of_quot_equiv f).symm
⟨↑j, dvd_of_mem_normalized_factors j.prop⟩, by { rw idea... | def | normalized_factors_equiv_of_quot_equiv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal_factors_equiv_of_quot_equiv",
"ideal_factors_equiv_of_quot_equiv_mem_normalized_factors_of_mem_normalized_factors",
"ideal_factors_equiv_of_quot_equiv_symm",
"inv_fun"
] | The bijection between the sets of normalized factors of I and J induced by a ring
isomorphism `f : R/I ≅ A/J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalized_factors_equiv_of_quot_equiv_symm (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
(normalized_factors_equiv_of_quot_equiv f hI hJ).symm =
normalized_factors_equiv_of_quot_equiv f.symm hJ hI | rfl | lemma | normalized_factors_equiv_of_quot_equiv_symm | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"normalized_factors_equiv_of_quot_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity (hI : I ≠ ⊥) (hJ : J ≠ ⊥)
(L : ideal R) (hL : L ∈ normalized_factors I) :
multiplicity ↑(normalized_factors_equiv_of_quot_equiv f hI hJ ⟨L, hL⟩) J = multiplicity L I | begin
rw [normalized_factors_equiv_of_quot_equiv, equiv.coe_fn_mk, subtype.coe_mk],
exact multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalized_factor hI hJ hL
(λ ⟨l, hl⟩ ⟨l', hl'⟩, ideal_factors_equiv_of_quot_equiv_is_dvd_iso f hl hl'),
end | lemma | normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"equiv.coe_fn_mk",
"ideal",
"ideal_factors_equiv_of_quot_equiv_is_dvd_iso",
"multiplicity",
"multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalized_factor",
"normalized_factors_equiv_of_quot_equiv",
"subtype.coe_mk"
] | The map `normalized_factors_equiv_of_quot_equiv` preserves multiplicities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.dimension_le_one.prime_le_prime_iff_eq (h : ring.dimension_le_one R)
{P Q : ideal R} [hP : P.is_prime] [hQ : Q.is_prime] (hP0 : P ≠ ⊥) :
P ≤ Q ↔ P = Q | ⟨(h P hP0 hP).eq_of_le hQ.ne_top, eq.le⟩ | lemma | ring.dimension_le_one.prime_le_prime_iff_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ring.dimension_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.coprime_of_no_prime_ge {I J : ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬ is_prime P) :
I ⊔ J = ⊤ | begin
by_contra hIJ,
obtain ⟨P, hP, hIJ⟩ := ideal.exists_le_maximal _ hIJ,
exact h P (le_trans le_sup_left hIJ) (le_trans le_sup_right hIJ) hP.is_prime
end | lemma | ideal.coprime_of_no_prime_ge | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"by_contra",
"ideal",
"ideal.exists_le_maximal",
"le_sup_left",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_prime.mul_mem_pow (I : ideal R) [hI : I.is_prime] {a b : R} {n : ℕ}
(h : a * b ∈ I^n) : a ∈ I ∨ b ∈ I^n | begin
cases n, { simp },
by_cases hI0 : I = ⊥, { simpa [pow_succ, hI0] using h },
simp only [← submodule.span_singleton_le_iff_mem, ideal.submodule_span_eq, ← ideal.dvd_iff_le,
← ideal.span_singleton_mul_span_singleton] at h ⊢,
by_cases ha : I ∣ span {a},
{ exact or.inl ha },
rw mul_comm at h,
exact o... | lemma | ideal.is_prime.mul_mem_pow | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.prime_iff_is_prime",
"ideal.span_singleton_mul_span_singleton",
"ideal.submodule_span_eq",
"mul_comm",
"pow_succ",
"prime.pow_dvd_of_dvd_mul_right",
"submodule.span_singleton_le_iff_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.count_normalized_factors_eq {p x : ideal R} [hp : p.is_prime] {n : ℕ}
(hle : x ≤ p^n) (hlt : ¬ (x ≤ p^(n+1))) :
(normalized_factors x).count p = n | count_normalized_factors_eq'
((ideal.is_prime_iff_bot_or_prime.mp hp).imp_right prime.irreducible)
(by { haveI : unique (ideal R)ˣ := ideal.unique_units, apply normalize_eq })
(by convert ideal.dvd_iff_le.mpr hle) (by convert mt ideal.le_of_dvd hlt) | lemma | ideal.count_normalized_factors_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.le_of_dvd",
"ideal.unique_units",
"normalize_eq",
"prime.irreducible",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.le_mul_of_no_prime_factors
{I J K : ideal R} (coprime : ∀ P, J ≤ P → K ≤ P → ¬ is_prime P) (hJ : I ≤ J) (hK : I ≤ K) :
I ≤ J * K | begin
simp only [← ideal.dvd_iff_le] at coprime hJ hK ⊢,
by_cases hJ0 : J = 0,
{ simpa only [hJ0, zero_mul] using hJ },
obtain ⟨I', rfl⟩ := hK,
rw mul_comm,
exact mul_dvd_mul_left K
(unique_factorization_monoid.dvd_of_dvd_mul_right_of_no_prime_factors hJ0
(λ P hPJ hPK, mt ideal.is_prime_of_prime (... | lemma | ideal.le_mul_of_no_prime_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.is_prime_of_prime",
"mul_comm",
"mul_dvd_mul_left",
"unique_factorization_monoid.dvd_of_dvd_mul_right_of_no_prime_factors",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.le_of_pow_le_prime {I P : ideal R} [hP : P.is_prime] {n : ℕ} (h : I^n ≤ P) : I ≤ P | begin
by_cases hP0 : P = ⊥,
{ simp only [hP0, le_bot_iff] at ⊢ h,
exact pow_eq_zero h },
rw ← ideal.dvd_iff_le at ⊢ h,
exact ((ideal.prime_iff_is_prime hP0).mpr hP).dvd_of_dvd_pow h
end | lemma | ideal.le_of_pow_le_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.prime_iff_is_prime",
"le_bot_iff",
"pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.pow_le_prime_iff {I P : ideal R} [hP : P.is_prime] {n : ℕ} (hn : n ≠ 0) :
I^n ≤ P ↔ I ≤ P | ⟨ideal.le_of_pow_le_prime, λ h, trans (ideal.pow_le_self hn) h⟩ | lemma | ideal.pow_le_prime_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.pow_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.prod_le_prime {ι : Type*} {s : finset ι} {f : ι → ideal R} {P : ideal R}
[hP : P.is_prime] :
∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P | begin
by_cases hP0 : P = ⊥,
{ simp only [hP0, le_bot_iff],
rw [← ideal.zero_eq_bot, finset.prod_eq_zero_iff] },
simp only [← ideal.dvd_iff_le],
exact ((ideal.prime_iff_is_prime hP0).mpr hP).dvd_finset_prod_iff _
end | lemma | ideal.prod_le_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"finset",
"finset.prod_eq_zero_iff",
"ideal",
"ideal.dvd_iff_le",
"ideal.prime_iff_is_prime",
"ideal.zero_eq_bot",
"le_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_dedekind_domain.inf_prime_pow_eq_prod {ι : Type*}
(s : finset ι) (f : ι → ideal R) (e : ι → ℕ)
(prime : ∀ i ∈ s, prime (f i)) (coprime : ∀ i j ∈ s, i ≠ j → f i ≠ f j) :
s.inf (λ i, f i ^ e i) = ∏ i in s, f i ^ e i | begin
letI := classical.dec_eq ι,
revert prime coprime,
refine s.induction _ _,
{ simp },
intros a s ha ih prime coprime,
specialize ih (λ i hi, prime i (finset.mem_insert_of_mem hi))
(λ i hi j hj, coprime i (finset.mem_insert_of_mem hi) j (finset.mem_insert_of_mem hj)),
rw [finset.inf_insert, finset.... | lemma | is_dedekind_domain.inf_prime_pow_eq_prod | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"classical.dec_eq",
"finset",
"finset.inf_insert",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_insert",
"ideal",
"ideal.is_prime_of_prime",
"ideal.le_mul_of_no_prime_factors",
"ideal.le_of_pow_le_prime",
"ideal.mul_le_inf",
"ih",
"inf_le_left",
"inf_le_right",
"ne_... | The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.quotient_equiv_pi_of_prod_eq {ι : Type*} [fintype ι]
(I : ideal R) (P : ι → ideal R) (e : ι → ℕ)
(prime : ∀ i, prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j) (prod_eq : (∏ i, P i ^ e i) = I) :
R ⧸ I ≃+* Π i, R ⧸ (P i ^ e i) | (ideal.quot_equiv_of_eq (by { simp only [← prod_eq, finset.inf_eq_infi, finset.mem_univ, cinfi_pos,
← is_dedekind_domain.inf_prime_pow_eq_prod _ _ _ (λ i _, prime i) (λ i _ j _, coprime i j)] }))
.trans $
ideal.quotient_inf_ring_equiv_pi_quotient _ (λ i j hij, ideal.coprime_of_no_prime_ge (begin
intros P hPi hP... | def | is_dedekind_domain.quotient_equiv_pi_of_prod_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"cinfi_pos",
"finset.inf_eq_infi",
"finset.mem_univ",
"fintype",
"ideal",
"ideal.coprime_of_no_prime_ge",
"ideal.is_prime_of_prime",
"ideal.le_of_pow_le_prime",
"ideal.quot_equiv_of_eq",
"ideal.quotient_inf_ring_equiv_pi_quotient",
"is_dedekind_domain.inf_prime_pow_eq_prod",
"ne_zero",
"prim... | **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)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.quotient_equiv_pi_factors {I : ideal R} (hI : I ≠ ⊥) :
R ⧸ I ≃+* Π (P : (factors I).to_finset), R ⧸ ((P : ideal R) ^ (factors I).count P) | is_dedekind_domain.quotient_equiv_pi_of_prod_eq _ _ _
(λ (P : (factors I).to_finset), prime_of_factor _ (multiset.mem_to_finset.mp P.prop))
(λ i j hij, subtype.coe_injective.ne hij)
(calc ∏ (P : (factors I).to_finset), (P : ideal R) ^ (factors I).count (P : ideal R)
= ∏ P in (factors I).to_finset, P ^ (fact... | def | is_dedekind_domain.quotient_equiv_pi_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"associated_iff_eq",
"finset.prod_multiset_map_count",
"ideal",
"ideal.unique_units",
"is_dedekind_domain.quotient_equiv_pi_of_prod_eq",
"multiset.map_id'"
] | **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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.quotient_equiv_pi_factors_mk {I : ideal R} (hI : I ≠ ⊥)
(x : R) : is_dedekind_domain.quotient_equiv_pi_factors hI (ideal.quotient.mk I x) =
λ P, ideal.quotient.mk _ x | rfl | lemma | is_dedekind_domain.quotient_equiv_pi_factors_mk | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.quotient.mk",
"is_dedekind_domain.quotient_equiv_pi_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient_mul_equiv_quotient_prod (I J : ideal R)
(coprime : I ⊔ J = ⊤) :
(R ⧸ (I * J)) ≃+* (R ⧸ I) × R ⧸ J | ring_equiv.trans
(ideal.quot_equiv_of_eq (inf_eq_mul_of_coprime coprime).symm)
(ideal.quotient_inf_equiv_quotient_prod I J coprime) | def | ideal.quotient_mul_equiv_quotient_prod | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.quot_equiv_of_eq",
"ideal.quotient_inf_equiv_quotient_prod",
"ring_equiv.trans"
] | **Chinese remainder theorem**, specialized to two ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq {ι : Type*} {s : finset ι}
(I : ideal R) (P : ι → ideal R) (e : ι → ℕ)
(prime : ∀ i ∈ s, prime (P i)) (coprime : ∀ (i j ∈ s), i ≠ j → P i ≠ P j)
(prod_eq : (∏ i in s, P i ^ e i) = I) :
R ⧸ I ≃+* Π (i : s), R ⧸ (P i ^ e i) | is_dedekind_domain.quotient_equiv_pi_of_prod_eq I (λ (i : s), P i) (λ (i : s), e i)
(λ i, prime i i.2)
(λ i j h, coprime i i.2 j j.2 (subtype.coe_injective.ne h))
(trans (finset.prod_coe_sort s (λ i, P i ^ e i)) prod_eq) | def | is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"finset",
"finset.prod_coe_sort",
"ideal",
"is_dedekind_domain.quotient_equiv_pi_of_prod_eq",
"prime"
] | **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i in s, P i ^ e i`, then `R ⧸ I` factors as `Π (i : s), R ⧸ (P i ^ e i)`.
This is a version of `is_dedekind_domain.quotient_equiv_pi_of_prod_eq` where we restrict
the product to a finite subset `s` of a potentially infinite indexing ty... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.exists_representative_mod_finset {ι : Type*} {s : finset ι}
(P : ι → ideal R) (e : ι → ℕ)
(prime : ∀ i ∈ s, prime (P i)) (coprime : ∀ (i j ∈ s), i ≠ j → P i ≠ P j)
(x : Π (i : s), R ⧸ (P i ^ e i)) :
∃ y, ∀ i (hi : i ∈ s), ideal.quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ | begin
let f := is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq _ P e prime coprime rfl,
obtain ⟨y, rfl⟩ := f.surjective x,
obtain ⟨z, rfl⟩ := ideal.quotient.mk_surjective y,
exact ⟨z, λ i hi, rfl⟩
end | lemma | is_dedekind_domain.exists_representative_mod_finset | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"finset",
"ideal",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"is_dedekind_domain.quotient_equiv_pi_of_finset_prod_eq",
"prime"
] | Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.exists_forall_sub_mem_ideal {ι : Type*} {s : finset ι}
(P : ι → ideal R) (e : ι → ℕ)
(prime : ∀ i ∈ s, prime (P i)) (coprime : ∀ (i j ∈ s), i ≠ j → P i ≠ P j)
(x : s → R) :
∃ y, ∀ i (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i | begin
obtain ⟨y, hy⟩ := is_dedekind_domain.exists_representative_mod_finset P e prime coprime
(λ i, ideal.quotient.mk _ (x i)),
exact ⟨y, λ i hi, ideal.quotient.eq.mp (hy i hi)⟩
end | lemma | is_dedekind_domain.exists_forall_sub_mem_ideal | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"finset",
"ideal",
"ideal.quotient.mk",
"is_dedekind_domain.exists_representative_mod_finset",
"prime"
] | Corollary of the Chinese remainder theorem: given elements `x i : R`,
we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_singleton_dvd_span_singleton_iff_dvd {a b : R} :
(ideal.span {a}) ∣ (ideal.span ({b} : set R)) ↔ a ∣ b | ⟨λ h, mem_span_singleton.mp (dvd_iff_le.mp h (mem_span_singleton.mpr (dvd_refl b))),
λ h, dvd_iff_le.mpr (λ d hd, mem_span_singleton.mpr (dvd_trans h (mem_span_singleton.mp hd)))⟩ | lemma | span_singleton_dvd_span_singleton_iff_dvd | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"dvd_refl",
"dvd_trans",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_span_mem_normalized_factors_of_mem_normalized_factors [normalization_monoid R]
[decidable_eq R] [decidable_eq (ideal R)] {a b : R} (ha : a ∈ normalized_factors b) :
ideal.span ({a} : set R) ∈ normalized_factors (ideal.span ({b} : set R)) | begin
by_cases hb : b = 0,
{ rw [ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalized_factors_zero],
rw [hb, normalized_factors_zero] at ha,
simpa only [multiset.not_mem_zero] },
{ suffices : prime (ideal.span ({a} : set R)),
{ obtain ⟨c, hc, hc'⟩ := exists_mem_normalized_factors_of_dvd _ this.... | lemma | singleton_span_mem_normalized_factors_of_mem_normalized_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"by_contra",
"ideal",
"ideal.span",
"multiset.not_mem_zero",
"ne_zero",
"normalization_monoid",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_eq_multiplicity_span [decidable_rel ((∣) : R → R → Prop)]
[decidable_rel ((∣) : ideal R → ideal R → Prop)] {a b : R} :
multiplicity (ideal.span {a}) (ideal.span ({b} : set R)) = multiplicity a b | begin
by_cases h : finite a b,
{ rw ← part_enat.coe_get (finite_iff_dom.mp h),
refine (multiplicity.unique
(show (ideal.span {a})^(((multiplicity a b).get h)) ∣ (ideal.span {b}), from _) _).symm ;
rw [ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd],
exact pow_multi... | lemma | multiplicity_eq_multiplicity_span | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"finite",
"ideal",
"ideal.span",
"ideal.span_singleton_pow",
"multiplicity",
"multiplicity.is_greatest",
"multiplicity.unique",
"part_enat.coe_get",
"part_enat.lt_coe_iff",
"part_enat.not_dom_iff_eq_top",
"span_singleton_dvd_span_singleton_iff_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalized_factors_equiv_span_normalized_factors {r : R} (hr : r ≠ 0) :
{d : R | d ∈ normalized_factors r} ≃
{I : ideal R | I ∈ normalized_factors (ideal.span ({r} : set R))} | equiv.of_bijective
(λ d, ⟨ideal.span {↑d}, singleton_span_mem_normalized_factors_of_mem_normalized_factors d.prop⟩)
begin
split,
{ rintros ⟨a, ha⟩ ⟨b, hb⟩ h,
rw [subtype.mk_eq_mk, ideal.span_singleton_eq_span_singleton, subtype.coe_mk,
subtype.coe_mk] at h,
exact subtype.mk_eq_mk.mpr (mem_normalized... | def | normalized_factors_equiv_span_normalized_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"dvd_refl",
"equiv.of_bijective",
"ideal",
"ideal.span",
"ideal.span_singleton_eq_span_singleton",
"ideal.span_singleton_generator",
"irreducible",
"ne_zero",
"singleton_span_mem_normalized_factors_of_mem_normalized_factors",
"submodule.is_principal.mem_iff_generator_dvd",
"submodule.is_principa... | The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
of `span {r}` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiplicity_normalized_factors_equiv_span_normalized_factors_eq_multiplicity {r d: R}
(hr : r ≠ 0) (hd : d ∈ normalized_factors r) :
multiplicity d r =
multiplicity (normalized_factors_equiv_span_normalized_factors hr ⟨d, hd⟩ : ideal R)
(ideal.span {r}) | by simp only [normalized_factors_equiv_span_normalized_factors, multiplicity_eq_multiplicity_span,
subtype.coe_mk, equiv.of_bijective_apply] | lemma | multiplicity_normalized_factors_equiv_span_normalized_factors_eq_multiplicity | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"ideal",
"ideal.span",
"multiplicity",
"multiplicity_eq_multiplicity_span",
"normalized_factors_equiv_span_normalized_factors",
"subtype.coe_mk"
] | The bijection `normalized_factors_equiv_span_normalized_factors` between the set of prime
factors of `r` and the set of prime factors of the ideal `⟨r⟩` preserves multiplicities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity
{r : R} (hr : r ≠ 0) (I : {I : ideal R | I ∈ normalized_factors (ideal.span ({r} : set R))}) :
multiplicity ((normalized_factors_equiv_span_normalized_factors hr).symm I : R) r =
multiplicity (I : ideal R) (ideal.span {r}) | begin
obtain ⟨x, hx⟩ := (normalized_factors_equiv_span_normalized_factors hr).surjective I,
obtain ⟨a, ha⟩ := x,
rw [hx.symm, equiv.symm_apply_apply, subtype.coe_mk,
multiplicity_normalized_factors_equiv_span_normalized_factors_eq_multiplicity hr ha, hx],
end | lemma | multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"equiv.symm_apply_apply",
"ideal",
"ideal.span",
"multiplicity",
"multiplicity_normalized_factors_equiv_span_normalized_factors_eq_multiplicity",
"normalized_factors_equiv_span_normalized_factors",
"subtype.coe_mk"
] | The bijection `normalized_factors_equiv_span_normalized_factors.symm` between the set of prime
factors of the ideal `⟨r⟩` and the set of prime factors of `r` preserves multiplicities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_closure.is_localization [is_separable K L] [no_zero_smul_divisors A L] :
is_localization (algebra.algebra_map_submonoid C A⁰) L | begin
haveI : is_domain C :=
(is_integral_closure.equiv A C L (integral_closure A L)).to_ring_equiv.is_domain
(integral_closure A L),
haveI : no_zero_smul_divisors A C := is_integral_closure.no_zero_smul_divisors A L,
refine ⟨_, λ z, _, λ x y, ⟨λ h, ⟨1, _⟩, _⟩⟩,
{ rintros ⟨_, x, hx, rfl⟩,
rw [is_u... | lemma | is_integral_closure.is_localization | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.algebra_map_submonoid",
"algebra_map",
"integral_closure",
"is_domain",
"is_integral.exists_multiple_integral_of_is_localization",
"is_integral_closure.equiv",
"is_integral_closure.no_zero_smul_divisors",
"is_localization",
"is_scalar_tower.algebra_map_apply",
"is_separable",
"is_separa... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.range_le_span_dual_basis [is_separable K L]
{ι : Type*} [fintype ι] [decidable_eq ι] (b : basis ι K L)
(hb_int : ∀ i, is_integral A (b i)) [is_integrally_closed A] :
((algebra.linear_map C L).restrict_scalars A).range ≤
submodule.span A (set.range $ (trace_form K L).dual_basis (trace_form_... | begin
let db := (trace_form K L).dual_basis (trace_form_nondegenerate K L) b,
rintros _ ⟨x, rfl⟩,
simp only [linear_map.coe_restrict_scalars_eq_coe, algebra.linear_map_apply],
have hx : is_integral A (algebra_map C L x) :=
(is_integral_closure.is_integral A L x).algebra_map,
rsuffices ⟨c, x_eq⟩ : ∃ (c : ι... | lemma | is_integral_closure.range_le_span_dual_basis | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.linear_map",
"algebra.linear_map_apply",
"algebra_map",
"basis",
"bilin_form.dual_basis_repr_apply",
"fintype",
"is_integral",
"is_integral_closure.is_integral",
"is_integral_mul",
"is_integrally_closed",
"is_localization.is_integer",
"is_scalar_tower.algebra_map_smul",
"is_separabl... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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