fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
(I ⊔ ∏ i ∈ s, J i) = ⊤ :=
Finset.prod_induction _ (fun J => I ⊔ J = ⊤)
(fun _ _ hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK)
(by simp_rw [one_eq_top, sup_top_eq]) h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | sup_prod_eq_top | null |
sup_multiset_prod_eq_top {s : Multiset (Ideal R)} (h : ∀ p ∈ s, I ⊔ p = ⊤) :
I ⊔ Multiset.prod s = ⊤ :=
Multiset.prod_induction (I ⊔ · = ⊤) s (fun _ _ hp hq ↦ (sup_mul_eq_of_coprime_left hp).trans hq)
(by simp only [one_eq_top, le_top, sup_of_le_right]) h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | sup_multiset_prod_eq_top | null |
sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
(I ⊔ ⨅ i ∈ s, J i) = ⊤ := by
rw [eq_top_iff, ← sup_prod_eq_top h, ← Finset.inf_eq_iInf]
grw [prod_le_inf] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | sup_iInf_eq_top | null |
prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(∏ i ∈ s, J i) ⊔ I = ⊤ := by rw [sup_comm, sup_prod_eq_top]; intro i hi; rw [sup_comm, h i hi] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | prod_sup_eq_top | null |
iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(⨅ i ∈ s, J i) ⊔ I = ⊤ := by rw [sup_comm, sup_iInf_eq_top]; intro i hi; rw [sup_comm, h i hi] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | iInf_sup_eq_top | null |
sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by
rw [← Finset.card_range n, ← Finset.prod_const]
exact sup_prod_eq_top fun _ _ => h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | sup_pow_eq_top | null |
pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by
rw [← Finset.card_range n, ← Finset.prod_const]
exact prod_sup_eq_top fun _ _ => h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | pow_sup_eq_top | null |
pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ :=
sup_pow_eq_top (pow_sup_eq_top h)
variable (I) in
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | pow_sup_pow_eq_top | null |
mul_top : I * ⊤ = I :=
Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | mul_top | null |
@[simp]
multiset_prod_eq_bot {R : Type*} [CommSemiring R] [IsDomain R] {s : Multiset (Ideal R)} :
s.prod = ⊥ ↔ ⊥ ∈ s :=
Multiset.prod_eq_zero_iff | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | multiset_prod_eq_bot | A product of ideals in an integral domain is zero if and only if one of the terms is zero. |
span_pair_mul_span_pair (w x y z : R) :
(span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by
simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | span_pair_mul_span_pair | null |
isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
simp only [one_eq_top, Submodule.mem_top]
· intro h
refine ⟨1, 1, ?_⟩
simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_iff_codisjoint | null |
isCoprime_of_isMaximal [I.IsMaximal] [J.IsMaximal] (ne : I ≠ J) : IsCoprime I J := by
rw [isCoprime_iff_codisjoint, isMaximal_def] at *
exact IsCoatom.codisjoint_of_ne ‹_› ‹_› ne | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_of_isMaximal | null |
isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by
rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_iff_add | null |
isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by
rw [← add_eq_one_iff, isCoprime_iff_add] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_iff_exists | null |
isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by
rw [isCoprime_iff_codisjoint, codisjoint_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_iff_sup_eq | null |
coprime_of_no_prime_ge {I J : Ideal R} (h : ∀ P, I ≤ P → J ≤ P → ¬IsPrime P) :
IsCoprime I J := by
rw [isCoprime_iff_sup_eq]
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.isPrime
open List in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | coprime_of_no_prime_ge | null |
isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by
rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
← isCoprime_iff_sup_eq]
simp | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_tfae | null |
_root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J :=
isCoprime_iff_codisjoint.mp h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | _root_.IsCoprime.codisjoint | null |
_root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | _root_.IsCoprime.add_eq | null |
_root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 :=
isCoprime_iff_exists.mp h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | _root_.IsCoprime.exists | null |
_root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | _root_.IsCoprime.sup_eq | null |
inf_eq_mul_of_isCoprime (coprime : IsCoprime I J) : I ⊓ J = I * J :=
(Ideal.mul_eq_inf_of_coprime coprime.sup_eq).symm | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | inf_eq_mul_of_isCoprime | null |
isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
mem_span_singleton]
constructor
· rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩
· rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_span_singleton_iff | null |
isCoprime_biInf {J : ι → Ideal R} {s : Finset ι}
(hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
classical
simp_rw [isCoprime_iff_add] at *
induction s using Finset.induction with
| empty =>
simp
| insert i s _ hs =>
rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
set K := ⨅ j ∈ s, J j
calc
1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
_ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one]
_ = (1+K)*I + K*J i := by ring
_ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isCoprime_biInf | null |
radical (I : Ideal R) : Ideal R where
carrier := { r | ∃ n : ℕ, r ^ n ∈ I }
zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩
add_mem' := fun {_ _} ⟨m, hxmi⟩ ⟨n, hyni⟩ =>
⟨m + n - 1, add_pow_add_pred_mem_of_pow_mem I hxmi hyni⟩
smul_mem' {r s} := fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical | The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. |
mem_radical_iff {r : R} : r ∈ I.radical ↔ ∃ n : ℕ, r ^ n ∈ I := Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | mem_radical_iff | null |
IsRadical (I : Ideal R) : Prop :=
I.radical ≤ I | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsRadical | An ideal is radical if it contains its radical. |
le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | le_radical | null |
radical_eq_iff : I.radical = I ↔ I.IsRadical := by
rw [le_antisymm_iff, and_iff_left le_radical, IsRadical]
alias ⟨_, IsRadical.radical⟩ := radical_eq_iff | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_eq_iff | An ideal is radical iff it is equal to its radical. |
isRadical_iff_pow_one_lt (k : ℕ) (hk : 1 < k) : I.IsRadical ↔ ∀ r, r ^ k ∈ I → r ∈ I :=
⟨fun h _r hr ↦ h ⟨k, hr⟩, fun h x ⟨n, hx⟩ ↦
k.pow_imp_self_of_one_lt hk _ (fun _ _ ↦ .inr ∘ I.smul_mem _) h n x hx⟩
variable (R) in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isRadical_iff_pow_one_lt | null |
radical_top : (radical ⊤ : Ideal R) = ⊤ :=
(eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_top | null |
radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩
variable (I) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_mono | null |
radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ =>
⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_isRadical | null |
radical_idem : radical (radical I) = radical I :=
(radical_isRadical I).radical
variable {I} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_idem | null |
IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J :=
⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsRadical.radical_le_iff | null |
radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J :=
(radical_isRadical J).radical_le_iff | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_le_radical_iff | null |
radical_eq_top : radical I = ⊤ ↔ I = ⊤ :=
⟨fun h =>
(eq_top_iff_one _).2 <|
let ⟨n, hn⟩ := (eq_top_iff_one _).1 h
@one_pow R _ n ▸ hn,
fun h => h.symm ▸ radical_top R⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_eq_top | null |
IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ =>
H.mem_of_pow_mem n hrni | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.isRadical | null |
IsPrime.radical (H : IsPrime I) : radical I = I :=
IsRadical.radical H.isRadical | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.radical | null |
mem_radical_of_pow_mem {I : Ideal R} {x : R} {m : ℕ} (hx : x ^ m ∈ radical I) :
x ∈ radical I :=
radical_idem I ▸ ⟨m, hx⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | mem_radical_of_pow_mem | null |
disjoint_powers_iff_notMem (y : R) (hI : I.IsRadical) :
Disjoint (Submonoid.powers y : Set R) ↑I ↔ y ∉ I.1 := by
refine ⟨fun h => Set.disjoint_left.1 h (Submonoid.mem_powers _),
fun h => disjoint_iff.mpr (eq_bot_iff.mpr ?_)⟩
rintro x ⟨⟨n, rfl⟩, hx'⟩
exact h (hI <| mem_radical_of_pow_mem <| le_radical hx')
@[deprecated (since := "2025-05-23")]
alias disjoint_powers_iff_not_mem := disjoint_powers_iff_notMem
variable (I J) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | disjoint_powers_iff_notMem | null |
radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) :=
le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <|
radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_sup | null |
radical_inf : radical (I ⊓ J) = radical I ⊓ radical J :=
le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))
fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm,
(pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩
variable {I J} in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_inf | null |
IsRadical.inf (hI : IsRadical I) (hJ : IsRadical J) : IsRadical (I ⊓ J) := by
rw [IsRadical, radical_inf]; exact inf_le_inf hI hJ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsRadical.inf | null |
isRadical_bot_iff :
(⊥ : Ideal R).IsRadical ↔ IsReduced R := by
simp only [IsRadical, SetLike.le_def, Ideal.mem_radical_iff, Ideal.mem_bot,
forall_exists_index, isReduced_iff, IsNilpotent] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isRadical_bot_iff | null |
isRadical_bot [IsReduced R] : (⊥ : Ideal R).IsRadical := by rwa [Ideal.isRadical_bot_iff] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isRadical_bot | null |
radicalInfTopHom : InfTopHom (Ideal R) (Ideal R) where
toFun := radical
map_inf' := radical_inf
map_top' := radical_top _
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radicalInfTopHom | `Ideal.radical` as an `InfTopHom`, bundling in that it distributes over `inf`. |
radicalInfTopHom_apply (I : Ideal R) : radicalInfTopHom I = radical I := rfl
open Finset in | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radicalInfTopHom_apply | null |
radical_finset_inf {ι} {s : Finset ι} {f : ι → Ideal R} {i : ι} (hi : i ∈ s)
(hs : ∀ ⦃y⦄, y ∈ s → (f y).radical = (f i).radical) :
(s.inf f).radical = (f i).radical := by
rw [← radicalInfTopHom_apply, map_finset_inf, ← Finset.inf'_eq_inf ⟨_, hi⟩]
exact Finset.inf'_eq_of_forall _ _ hs | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_finset_inf | null |
radical_iInf_le {ι} (I : ι → Ideal R) : radical (⨅ i, I i) ≤ ⨅ i, radical (I i) :=
le_iInf fun _ ↦ radical_mono (iInf_le _ _) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_iInf_le | The reverse inclusion does not hold for e.g. `I := fun n : ℕ ↦ Ideal.span {(2 ^ n : ℤ)}`. |
isRadical_iInf {ι} (I : ι → Ideal R) (hI : ∀ i, IsRadical (I i)) : IsRadical (⨅ i, I i) :=
(radical_iInf_le I).trans (iInf_mono hI) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isRadical_iInf | null |
radical_mul : radical (I * J) = radical I ⊓ radical J := by
refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩
have := radical_mono <| @mul_le_inf _ _ I J
simp_rw [radical_inf I J] at this
assumption
variable {I J} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_mul | null |
IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J :=
IsRadical.radical_le_iff hJ.isRadical | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.radical_le_iff | null |
radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } :=
le_antisymm (le_sInf fun _ hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦
by_contradiction fun hri ↦
let ⟨m, hIm, hm⟩ :=
zorn_le_nonempty₀ { K : Ideal R | r ∉ radical K }
(fun c hc hcc y hyc =>
⟨sSup c, fun ⟨n, hrnc⟩ =>
let ⟨_, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc
hc hyc ⟨n, hrny⟩,
fun _ => le_sSup⟩)
I hri
have hrm : r ∉ radical m := hm.prop
have : ∀ x ∉ m, r ∈ radical (m ⊔ span {x}) := fun x hxm =>
by_contradiction fun hrmx => hxm <| by
rw [hm.eq_of_le hrmx le_sup_left]
exact Submodule.mem_sup_right <| mem_span_singleton_self x
have : IsPrime m :=
⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym =>
or_iff_not_imp_left.2 fun hxm =>
by_contradiction fun hym =>
let ⟨n, hrn⟩ := this _ hxm
let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn
let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq
let ⟨k, hrk⟩ := this _ hym
let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk
let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg
hrm
⟨n + k, by
rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x),
mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]
refine
m.add_mem (m.mul_mem_right _ hpm)
(m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩
hrm <|
this.radical.symm ▸ (sInf_le ⟨hIm, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_eq_sInf | null |
isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] :
(⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isRadical_bot_of_noZeroDivisors | null |
radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] :
radical (⊥ : Ideal R) = ⊥ :=
eq_bot_iff.2 isRadical_bot_of_noZeroDivisors | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_bot_of_noZeroDivisors | null |
top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ :=
Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | top_pow | null |
pow_eq_top_iff {n : ℕ} :
I ^ n = ⊤ ↔ I = ⊤ ∨ n = 0 := by
refine ⟨fun h ↦ or_iff_not_imp_right.mpr
fun hn ↦ (eq_top_iff_one _).mpr <| pow_le_self hn <| (eq_top_iff_one _).mp h, ?_⟩
rintro (h | h)
· rw [h, top_pow]
· rw [h, pow_zero, one_eq_top] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | pow_eq_top_iff | null |
natCast_eq_top {n : ℕ} (hn : n ≠ 0) : (n : Ideal R) = ⊤ :=
natCast_eq_one hn |>.trans one_eq_top | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | natCast_eq_top | null |
ofNat_eq_top {n : ℕ} [n.AtLeastTwo] : (ofNat(n) : Ideal R) = ⊤ :=
ofNat_eq_one.trans one_eq_top
variable (I) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | ofNat_eq_top | `3 : Ideal R` is *not* the ideal generated by 3 (which would be spelt
`Ideal.span {3}`), it is simply `1 + 1 + 1 = ⊤`. |
radical_pow : ∀ {n}, n ≠ 0 → radical (I ^ n) = radical I
| 1, _ => by simp
| n + 2, _ => by rw [pow_succ, radical_mul, radical_pow n.succ_ne_zero, inf_idem] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | radical_pow | null |
IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by
rw [or_comm, Ideal.mul_le]
simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.mul_le | null |
IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P :=
⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.inf_le | null |
IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P :=
s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.multiset_prod_le | null |
IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R}
(hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.multiset_prod_map_le | null |
IsPrime.multiset_prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime) (s : Multiset R) :
s.prod ∈ I ↔ ∃ p ∈ s, p ∈ I := by
simpa [span_singleton_le_iff_mem] using (hI.multiset_prod_map_le (span {·})) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.multiset_prod_mem_iff_exists_mem | null |
IsPrime.pow_le_iff {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (hn : n ≠ 0) :
I ^ n ≤ P ↔ I ≤ P := by
have h : (Multiset.replicate n I).prod ≤ P ↔ _ := hP.multiset_prod_le
simp_rw [Multiset.prod_replicate, Multiset.mem_replicate, ne_eq, hn, not_false_eq_true,
true_and, exists_eq_left] at h
exact h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.pow_le_iff | null |
IsPrime.le_of_pow_le {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
I ≤ P := by
by_cases hn : n = 0
· rw [hn, pow_zero, one_eq_top] at h
exact fun ⦃_⦄ _ ↦ h Submodule.mem_top
· exact (pow_le_iff hn).mp h | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.le_of_pow_le | null |
IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) :
s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
hp.multiset_prod_map_le f | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.prod_le | null |
IsPrime.prod_mem_iff {s : Finset ι} {x : ι → R} {p : Ideal R} [hp : p.IsPrime] :
∏ i ∈ s, x i ∈ p ↔ ∃ i ∈ s, x i ∈ p := by
simp_rw [← span_singleton_le_iff_mem, ← prod_span_singleton]
exact hp.prod_le | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.prod_mem_iff | The product of a finite number of elements in the commutative semiring `R` lies in the
prime ideal `p` if and only if at least one of those elements is in `p`. |
IsPrime.prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime) (s : Finset R) :
s.prod (fun x ↦ x) ∈ I ↔ ∃ p ∈ s, p ∈ I := by
rw [Finset.prod_eq_multiset_prod, Multiset.map_id']
exact hI.multiset_prod_mem_iff_exists_mem s.val | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.prod_mem_iff_exists_mem | null |
IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) :
s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | IsPrime.inf_le' | null |
subset_union {R : Type u} [Ring R] {I J K : Ideal R} :
(I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K :=
AddSubgroupClass.subset_union | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | subset_union | null |
subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by
suffices
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from
⟨this, fun h =>
Or.casesOn h
(fun h =>
Set.Subset.trans h <|
Set.Subset.trans Set.subset_union_left Set.subset_union_left)
fun h =>
Or.casesOn h
(fun h =>
Set.Subset.trans h <|
Set.Subset.trans Set.subset_union_right Set.subset_union_left)
fun ⟨i, his, hi⟩ => by
refine Set.Subset.trans hi <| Set.Subset.trans ?_ Set.subset_union_right
exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩
generalize hn : s.card = n; intro h
induction n generalizing a b s with
| zero =>
clear hp
rw [Finset.card_eq_zero] at hn
subst hn
rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
simpa only [exists_prop, Finset.notMem_empty, false_and, exists_false, or_false]
| succ n ih =>
classical
replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
Finset.card_eq_succ.1 hn
rcases hn with ⟨i, t, hit, rfl, hn⟩
replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp
by_cases Ht : ∃ j ∈ t, f j ≤ f i
· obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht
obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t :=
⟨t.erase j, t.notMem_erase j, Finset.insert_erase hjt⟩
have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by
rw [Finset.forall_mem_insert] at hp ⊢
exact ⟨hp.1, hp.2.2⟩
have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit
have hn' : (insert i u).card = n := by
rwa [Finset.card_insert_of_notMem] at hn ⊢
exacts [hiu, hju]
have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by
rw [Finset.coe_insert] at h ⊢
rw [Finset.coe_insert] at h
simp only [Set.biUnion_insert] at h ⊢
rw [← Set.union_assoc (f i : Set R),
Set.union_eq_self_of_subset_right hfji] at h
exact h
specialize ih hp' hn' h'
... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | subset_union_prime' | null |
@[stacks 00DS]
subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι)
(hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i :=
suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by
have aux := fun h => (bex_def.2 <| this h)
simp_rw [exists_prop] at aux
refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩
apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his)
fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by
classical
by_cases has : a ∈ s
· obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s :=
⟨s.erase a, Finset.notMem_erase a s, Finset.insert_erase has⟩
by_cases hbt : b ∈ t
· obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t :=
⟨t.erase b, Finset.notMem_erase b t, Finset.insert_erase hbt⟩
have hp' : ∀ i ∈ u, IsPrime (f i) := by
intro i hiu
refine hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) ?_ ?_ <;>
rintro rfl <;>
solve_by_elim only [Finset.mem_insert_of_mem, *]
rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ←
Set.union_assoc, subset_union_prime' hp'] at h
rwa [Finset.exists_mem_insert, Finset.exists_mem_insert]
· have hp' : ∀ j ∈ t, IsPrime (f j) := by
intro j hj
refine hp j (Finset.mem_insert_of_mem hj) ?_ ?_ <;> rintro rfl <;>
solve_by_elim only [Finset.mem_insert_of_mem, *]
rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R),
subset_union_prime' hp', ← or_assoc, or_self_iff] at h
rwa [Finset.exists_mem_insert]
· by_cases hbs : b ∈ s
· obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s :=
⟨s.erase b, Finset.notMem_erase b s, Finset.insert_erase hbs⟩
have hp' : ∀ j ∈ t, IsPrime (f j) := by
intro j hj
refine hp j (Finset.mem_insert_of_mem hj) ?_ ?_ <;> rintro rfl <;>
solve_by_elim only [Finset.mem_insert_of_mem, *]
rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R),
subset_union_prime' hp', ← or_assoc, or_self_iff] at h
rwa [Finset.exists_mem_insert]
rcases s.eq_empty_or_nonempty with hse | hsne
· subst hse
rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h
have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)
exact absurd h this
· obtain ⟨i, his⟩ := hsne
obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s :=
⟨s.erase i, Finset.notMem_erase i s, Finset.insert_erase his⟩
have hp' : ∀ j ∈ t, IsPrime (f j) := by
... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | subset_union_prime | Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Matsumura Ex.1.6. |
subset_union_prime_finite {R ι : Type*} [CommRing R] {s : Set ι}
(hs : s.Finite) {f : ι → Ideal R} (a b : ι)
(hp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime) {I : Ideal R} :
((I : Set R) ⊆ ⋃ i ∈ s, f i) ↔ ∃ i ∈ s, I ≤ f i := by
rcases Set.Finite.exists_finset hs with ⟨t, ht⟩
have heq : ⋃ i ∈ s, f i = ⋃ i ∈ t, (f i : Set R) := by
ext
simpa using exists_congr (fun i ↦ (and_congr_left fun a ↦ ht i).symm)
have hmem_union : ((I : Set R) ⊆ ⋃ i ∈ s, f i) ↔ ((I : Set R) ⊆ ⋃ i ∈ (t : Set ι), f i) :=
(congrArg _ heq).to_iff
rw [hmem_union, Ideal.subset_union_prime a b (fun i hin ↦ hp i ((ht i).mp hin))]
exact exists_congr (fun i ↦ and_congr_left fun _ ↦ ht i) | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | subset_union_prime_finite | Another version of prime avoidance using `Set.Finite` instead of `Finset`. |
le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I
| ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | le_of_dvd | If `I` divides `J`, then `I` contains `J`.
In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. |
dvd_bot {I : Ideal R} : I ∣ ⊥ :=
dvd_zero I | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | dvd_bot | null |
@[simp high]
isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ :=
isUnit_iff_dvd_one.trans
((@one_eq_top R _).symm ▸
⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | isUnit_iff | See also `isUnit_iff_eq_one`. |
uniqueUnits : Unique (Ideal R)ˣ where
default := 1
uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | uniqueUnits | null |
noncomputable finsuppTotal : (ι →₀ I) →ₗ[R] M :=
(Finsupp.linearCombination R v).comp (Finsupp.mapRange.linearMap I.subtype)
variable {ι M v} | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | finsuppTotal | A variant of `Finsupp.linearCombination` that takes in vectors valued in `I`. |
finsuppTotal_apply (f : ι →₀ I) :
finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i := by
dsimp [finsuppTotal]
rw [Finsupp.linearCombination_apply, Finsupp.sum_mapRange_index]
exact fun _ => zero_smul _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | finsuppTotal_apply | null |
finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i := by
rw [finsuppTotal_apply, Finsupp.sum_fintype]
exact fun _ => zero_smul _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | finsuppTotal_apply_eq_of_fintype | null |
range_finsuppTotal :
LinearMap.range (finsuppTotal ι M I v) = I • Submodule.span R (Set.range v) := by
ext
rw [Submodule.mem_ideal_smul_span_iff_exists_sum]
refine ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, ?_⟩
rintro ⟨a, ha, rfl⟩
classical
refine ⟨a.mapRange (fun r => if h : r ∈ I then ⟨r, h⟩ else 0)
(by simp only [Submodule.zero_mem, ↓reduceDIte]; rfl), ?_⟩
rw [finsuppTotal_apply, Finsupp.sum_mapRange_index]
· apply Finsupp.sum_congr
intro i _
rw [dif_pos (ha i)]
· exact fun _ => zero_smul _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | range_finsuppTotal | null |
Finsupp.mem_ideal_span_range_iff_exists_finsupp {x : R} {v : α → R} :
x ∈ Ideal.span (Set.range v) ↔ ∃ c : α →₀ R, (c.sum fun i a => a * v i) = x :=
Finsupp.mem_span_range_iff_exists_finsupp | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Finsupp.mem_ideal_span_range_iff_exists_finsupp | null |
Ideal.mem_span_range_iff_exists_fun [Fintype α] {x : R} {v : α → R} :
x ∈ Ideal.span (Set.range v) ↔ ∃ c : α → R, ∑ i, c i * v i = x :=
Submodule.mem_span_range_iff_exists_fun _
@[deprecated (since := "2025-04-02")] alias mem_ideal_span_range_iff_exists_fun :=
Ideal.mem_span_range_iff_exists_fun | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Ideal.mem_span_range_iff_exists_fun | An element `x` lies in the span of `v` iff it can be written as sum `∑ cᵢ • vᵢ = x`. |
Associates.mk_ne_zero' {R : Type*} [CommSemiring R] {r : R} :
Associates.mk (Ideal.span {r} : Ideal R) ≠ 0 ↔ r ≠ 0 := by
rw [Associates.mk_ne_zero, Ideal.zero_eq_bot, Ne, Ideal.span_singleton_eq_bot]
open scoped nonZeroDivisors in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Associates.mk_ne_zero' | null |
Ideal.span_singleton_nonZeroDivisors {R : Type*} [CommSemiring R] [NoZeroDivisors R]
{r : R} : span {r} ∈ (Ideal R)⁰ ↔ r ∈ R⁰ := by
cases subsingleton_or_nontrivial R
· simp_rw [← nonZeroDivisorsRight_eq_nonZeroDivisors]
exact ⟨fun _ _ _ ↦ Subsingleton.eq_zero _, fun _ _ _ ↦ Subsingleton.eq_zero _⟩
· rw [mem_nonZeroDivisors_iff_ne_zero, mem_nonZeroDivisors_iff_ne_zero, ne_eq, zero_eq_bot,
span_singleton_eq_bot] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Ideal.span_singleton_nonZeroDivisors | null |
Ideal.primeCompl_le_nonZeroDivisors {R : Type*} [CommSemiring R] [NoZeroDivisors R]
(P : Ideal R) [P.IsPrime] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Ideal.primeCompl_le_nonZeroDivisors | null |
moduleSubmodule : Module (Ideal R) (Submodule R M) where
smul_add := smul_sup
add_smul := sup_smul
mul_smul := Submodule.mul_smul
one_smul := by simp
zero_smul := bot_smul
smul_zero := smul_bot | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | moduleSubmodule | null |
span_smul_eq
(s : Set R) (N : Submodule R M) :
Ideal.span s • N = s • N := by
rw [← coe_set_smul, coe_span_smul]
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | span_smul_eq | null |
set_smul_top_eq_span (s : Set R) :
s • ⊤ = Ideal.span s :=
(span_smul_eq s ⊤).symm.trans (Ideal.span s).mul_top | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | set_smul_top_eq_span | null |
smul_le_span (s : Set R) (I : Ideal R) : s • I ≤ Ideal.span s := by
simp [← Submodule.set_smul_top_eq_span, smul_le_smul_left]
variable {A B} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
open Submodule | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | smul_le_span | null |
algebraIdeal : Algebra (Ideal R) (Submodule R A) where
__ := moduleSubmodule
algebraMap :=
{ toFun := map (Algebra.linearMap R A)
map_one' := by
rw [one_eq_span, map_span, Set.image_singleton, Algebra.linearMap_apply, map_one, one_eq_span]
map_mul' := (Submodule.map_mul · · <| Algebra.ofId R A)
map_zero' := map_bot _
map_add' := (map_sup · · _) }
commutes' I M := mul_comm_of_commute <| by rintro _ ⟨r, _, rfl⟩ a _; apply Algebra.commutes
smul_def' I M := le_antisymm (smul_le.mpr fun r hr a ha ↦ by
rw [Algebra.smul_def]; exact Submodule.mul_mem_mul ⟨r, hr, rfl⟩ ha) (Submodule.mul_le.mpr <| by
rintro _ ⟨r, hr, rfl⟩ a ha; rw [Algebra.linearMap_apply, ← Algebra.smul_def]
exact Submodule.smul_mem_smul hr ha) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | algebraIdeal | null |
@[simps!] mapAlgHom (f : A →ₐ[R] B) : Submodule R A →ₐ[Ideal R] Submodule R B where
__ := mapHom f
commutes' I := (map_comp _ _ I).symm.trans (congr_arg (map · I) <| LinearMap.ext f.commutes) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | mapAlgHom | `Submonoid.map` as an `AlgHom`, when applied to an `AlgHom`. |
@[simps!] mapAlgEquiv (f : A ≃ₐ[R] B) : Submodule R A ≃ₐ[Ideal R] Submodule R B where
__ := mapAlgHom f
invFun := mapAlgHom f.symm
left_inv I := (map_comp _ _ I).symm.trans <|
(congr_arg (map · I) <| LinearMap.ext (f.left_inv ·)).trans (map_id I)
right_inv I := (map_comp _ _ I).symm.trans <|
(congr_arg (map · I) <| LinearMap.ext (f.right_inv ·)).trans (map_id I) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | mapAlgEquiv | null |
Ideal.exists_subset_radical_span_sup_of_subset_radical_sup {R : Type*} [CommSemiring R]
(s : Set R) (I J : Ideal R) (hs : s ⊆ (I ⊔ J).radical) :
∃ (t : s → R), Set.range t ⊆ I ∧ s ⊆ (span (Set.range t) ⊔ J).radical := by
replace hs : ∀ z : s, ∃ (m : ℕ) (a b : R) (ha : a ∈ I) (hb : b ∈ J), a + b = z ^ m := by
rintro ⟨z, hzs⟩
simp only [Ideal.radical, Submodule.mem_sup] at hs
obtain ⟨m, y, hyq, b, hb, hy⟩ := hs hzs
exact ⟨m, y, b, hyq, hb, hy⟩
choose m a b ha hb heq using hs
refine ⟨a, by rwa [Set.range_subset_iff], fun z hz ↦ ⟨m ⟨z, hz⟩, heq ⟨z, hz⟩ ▸ ?_⟩⟩
exact Ideal.add_mem _ (mem_sup_left (subset_span ⟨⟨z, hz⟩, rfl⟩)) (mem_sup_right <| hb _)
@[deprecated (since := "2025-05-13")]
alias Ideal.exists_subset_radical_span_sup_span_of_subset_radical_sup :=
Ideal.exists_subset_radical_span_sup_of_subset_radical_sup | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.RingTheory.Nilpotent.Defs",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic"
] | Mathlib/RingTheory/Ideal/Operations.lean | Ideal.exists_subset_radical_span_sup_of_subset_radical_sup | null |
comap_eq_of_scalar_tower_quotient [Algebra R S] [Algebra (R ⧸ p) (S ⧸ P)]
[IsScalarTower R (R ⧸ p) (S ⧸ P)] (h : Function.Injective (algebraMap (R ⧸ p) (S ⧸ P))) :
comap (algebraMap R S) P = p := by
ext x
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,
IsScalarTower.algebraMap_apply R (R ⧸ p) (S ⧸ P), Quotient.algebraMap_eq]
constructor
· intro hx
exact (injective_iff_map_eq_zero (algebraMap (R ⧸ p) (S ⧸ P))).mp h _ hx
· intro hx
rw [hx, RingHom.map_zero]
variable [Algebra R S] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Quotient.Operations"
] | Mathlib/RingTheory/Ideal/Over.lean | comap_eq_of_scalar_tower_quotient | If there is an injective map `R/p → S/P` such that following diagram commutes:
```
R → S
↓ ↓
R/p → S/P
```
then `P` lies over `p`. |
Quotient.algebraQuotientMapQuotient : Algebra (R ⧸ p) (S ⧸ map (algebraMap R S) p) :=
Ideal.Quotient.algebraQuotientOfLEComap le_comap_map
@[simp] | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Group.Subgroup.Actions",
"Mathlib.RingTheory.Ideal.Pointwise",
"Mathlib.RingTheory.Ideal.Quotient.Operations"
] | Mathlib/RingTheory/Ideal/Over.lean | Quotient.algebraQuotientMapQuotient | `R / p` has a canonical map to `S / pS`. |
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