ring_theory.ideal.operations

source

@[instance]

def submodule.has_scalar' {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] :

has_scalar (ideal R) (submodule R M)

Equations

submodule.has_scalar' = {smul := λ (I : ideal R) (N : submodule R M), ⨆ (r : ↥I), ↑r • N}

source

def submodule.annihilator {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (N : submodule R M) :

ideal R

N.annihilator is the ideal of all elements r : R such that r • N = 0.

Equations

N.annihilator = (linear_map.lsmul R ↥N).ker

source

def submodule.colon {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (N P : submodule R M) :

ideal R

N.colon P is the ideal of all elements r : R such that r • P ⊆ N.

Equations

N.colon P = (submodule.map N.mkq P).annihilator

source

theorem submodule.mem_annihilator {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N : submodule R M} {r : R} :

r ∈ N.annihilator ↔ ∀ (n : M), n ∈ N → r • n = 0

source

theorem submodule.mem_annihilator' {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N : submodule R M} {r : R} :

r ∈ N.annihilator ↔ N ≤ submodule.comap (r • linear_map.id) ⊥

source

theorem submodule.annihilator_bot {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] :

⊥.annihilator = ⊤

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theorem submodule.annihilator_eq_top_iff {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N : submodule R M} :

N.annihilator = ⊤ ↔ N = ⊥

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theorem submodule.annihilator_mono {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N P : submodule R M} (h : N ≤ P) :

P.annihilator ≤ N.annihilator

source

theorem submodule.annihilator_supr {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (ι : Sort w) (f : ι → submodule R M) :

(⨆ (i : ι), f i).annihilator = ⨅ (i : ι), (f i).annihilator

source

theorem submodule.mem_colon {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N P : submodule R M} {r : R} :

r ∈ N.colon P ↔ ∀ (p : M), p ∈ P → r • p ∈ N

source

theorem submodule.mem_colon' {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N P : submodule R M} {r : R} :

r ∈ N.colon P ↔ P ≤ submodule.comap (r • linear_map.id) N

source

theorem submodule.colon_mono {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {N₁ N₂ P₁ P₂ : submodule R M} (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) :

N₁.colon P₂ ≤ N₂.colon P₁

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theorem submodule.infi_colon_supr {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (ι₁ : Sort w) (f : ι₁ → submodule R M) (ι₂ : Sort x) (g : ι₂ → submodule R M) :

(⨅ (i : ι₁), f i).colon (⨆ (j : ι₂), g j) = ⨅ (i : ι₁) (j : ι₂), (f i).colon (g j)

source

theorem submodule.smul_mem_smul {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {N : submodule R M} {r : R} {n : M} (hr : r ∈ I) (hn : n ∈ N) :

r • n ∈ I • N

source

theorem submodule.smul_le {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {N P : submodule R M} :

I • N ≤ P ↔ ∀ (r : R), r ∈ I → ∀ (n : M), n ∈ N → r • n ∈ P

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theorem submodule.smul_induction_on {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {N : submodule R M} {p : M → Prop} {x : M} (H : x ∈ I • N) (Hb : ∀ (r : R), r ∈ I → ∀ (n : M), n ∈ N → p (r • n)) (H0 : p 0) (H1 : ∀ (x y : M), p x → p y → p (x + y)) (H2 : ∀ (c : R) (n : M), p n → p (c • n)) :

p x

source

theorem submodule.mem_smul_span_singleton {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {m x : M} :

x ∈ I • submodule.span R {m} ↔ ∃ (y : R) (H : y ∈ I), y • m = x

source

theorem submodule.smul_le_right {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {N : submodule R M} :

I • N ≤ N

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theorem submodule.smul_mono {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I J : ideal R} {N P : submodule R M} (hij : I ≤ J) (hnp : N ≤ P) :

I • N ≤ J • P

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theorem submodule.smul_mono_left {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I J : ideal R} {N : submodule R M} (h : I ≤ J) :

I • N ≤ J • N

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theorem submodule.smul_mono_right {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] {I : ideal R} {N P : submodule R M} (h : N ≤ P) :

I • N ≤ I • P

source

@[simp]

theorem submodule.annihilator_smul {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (N : submodule R M) :

N.annihilator • N = ⊥

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@[simp]

theorem submodule.annihilator_mul {R : Type u} [comm_ring R] (I : ideal R) :

(submodule.annihilator I) * I = ⊥

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@[simp]

theorem submodule.mul_annihilator {R : Type u} [comm_ring R] (I : ideal R) :

I * submodule.annihilator I = ⊥

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@[simp]

theorem submodule.smul_bot {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (I : ideal R) :

I • ⊥ = ⊥

source

@[simp]

theorem submodule.bot_smul {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (N : submodule R M) :

⊥ • N = ⊥

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@[simp]

theorem submodule.top_smul {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (N : submodule R M) :

⊤ • N = N

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theorem submodule.smul_sup {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (I : ideal R) (N P : submodule R M) :

I • (N ⊔ P) = I • N ⊔ I • P

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theorem submodule.sup_smul {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (I J : ideal R) (N : submodule R M) :

(I ⊔ J) • N = I • N ⊔ J • N

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theorem submodule.smul_assoc {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (I J : ideal R) (N : submodule R M) :

(I • J) • N = I • J • N

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theorem submodule.span_smul_span {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (S : set R) (T : set M) :

ideal.span S • submodule.span R T = submodule.span R (⋃ (s : R) (H : s ∈ S) (t : M) (H : t ∈ T), {s • t})

source

theorem submodule.map_smul'' {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] (I : ideal R) (N : submodule R M) {M' : Type w} [add_comm_group M'] [module R M'] (f : M →ₗ[R] M') :

submodule.map f (I • N) = I • submodule.map f N

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theorem ideal.exists_sub_one_mem_and_mem {R : Type u} [comm_ring R] {ι : Type v} (s : finset ι) {f : ι → ideal R} (hf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤) (i : ι) (his : i ∈ s) :

∃ (r : R), r - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → r ∈ f j

source

theorem ideal.exists_sub_mem {R : Type u} [comm_ring R] {ι : Type v} [fintype ι] {f : ι → ideal R} (hf : ∀ (i j : ι), i ≠ j → f i ⊔ f j = ⊤) (g : ι → R) :

∃ (r : R), ∀ (i : ι), r - g i ∈ f i

source

def ideal.quotient_inf_to_pi_quotient {R : Type u} [comm_ring R] {ι : Type v} (f : ι → ideal R) :

(⨅ (i : ι), f i).quotient →+* Π (i : ι), (f i).quotient

The homomorphism from R/(⋂ i, f i) to ∏ i, (R / f i) featured in the Chinese Remainder Theorem. It is bijective if the ideals f i are comaximal.

Equations

ideal.quotient_inf_to_pi_quotient f = ideal.quotient.lift (⨅ (i : ι), f i) (pi.ring_hom (λ (i : ι), ideal.quotient.mk (f i))) _

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theorem ideal.quotient_inf_to_pi_quotient_bijective {R : Type u} [comm_ring R] {ι : Type v} [fintype ι] {f : ι → ideal R} (hf : ∀ (i j : ι), i ≠ j → f i ⊔ f j = ⊤) :

function.bijective ⇑(ideal.quotient_inf_to_pi_quotient f)

source

def ideal.quotient_inf_ring_equiv_pi_quotient {R : Type u} [comm_ring R] {ι : Type v} [fintype ι] (f : ι → ideal R) (hf : ∀ (i j : ι), i ≠ j → f i ⊔ f j = ⊤) :

(⨅ (i : ι), f i).quotient ≃+* Π (i : ι), (f i).quotient

Chinese Remainder Theorem. Eisenbud Ex.2.6. Similar to Atiyah-Macdonald 1.10 and Stacks 00DT

Equations

ideal.quotient_inf_ring_equiv_pi_quotient f hf = {to_fun := (equiv.of_bijective ⇑(ideal.quotient_inf_to_pi_quotient (λ (i : ι), f i)) _).to_fun, inv_fun := (equiv.of_bijective ⇑(ideal.quotient_inf_to_pi_quotient (λ (i : ι), f i)) _).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[instance]

def ideal.has_mul {R : Type u} [comm_ring R] :

has_mul (ideal R)

Equations

ideal.has_mul = {mul := has_scalar.smul submodule.has_scalar'}

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@[simp]

theorem ideal.add_eq_sup {R : Type u} [comm_ring R] {I J : ideal R} :

I + J = I ⊔ J

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@[simp]

theorem ideal.zero_eq_bot {R : Type u} [comm_ring R] :

0 = ⊥

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@[simp]

theorem ideal.one_eq_top {R : Type u} [comm_ring R] :

1 = ⊤

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theorem ideal.mul_mem_mul {R : Type u} [comm_ring R] {I J : ideal R} {r s : R} (hr : r ∈ I) (hs : s ∈ J) :

r * s ∈ I * J

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theorem ideal.mul_mem_mul_rev {R : Type u} [comm_ring R] {I J : ideal R} {r s : R} (hr : r ∈ I) (hs : s ∈ J) :

s * r ∈ I * J

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theorem ideal.pow_mem_pow {R : Type u} [comm_ring R] {I : ideal R} {x : R} (hx : x ∈ I) (n : ℕ) :

x ^ n ∈ I ^ n

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theorem ideal.mul_le {R : Type u} [comm_ring R] {I J K : ideal R} :

I * J ≤ K ↔ ∀ (r : R), r ∈ I → ∀ (s : R), s ∈ J → r * s ∈ K

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theorem ideal.mul_le_left {R : Type u} [comm_ring R] {I J : ideal R} :

I * J ≤ J

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theorem ideal.mul_le_right {R : Type u} [comm_ring R] {I J : ideal R} :

I * J ≤ I

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@[simp]

theorem ideal.sup_mul_right_self {R : Type u} [comm_ring R] {I J : ideal R} :

I ⊔ I * J = I

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@[simp]

theorem ideal.sup_mul_left_self {R : Type u} [comm_ring R] {I J : ideal R} :

I ⊔ J * I = I

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@[simp]

theorem ideal.mul_right_self_sup {R : Type u} [comm_ring R] {I J : ideal R} :

I * J ⊔ I = I

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@[simp]

theorem ideal.mul_left_self_sup {R : Type u} [comm_ring R] {I J : ideal R} :

J * I ⊔ I = I

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theorem ideal.mul_comm {R : Type u} [comm_ring R] (I J : ideal R) :

I * J = J * I

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theorem ideal.mul_assoc {R : Type u} [comm_ring R] (I J K : ideal R) :

(I * J) * K = I * J * K

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theorem ideal.span_mul_span {R : Type u} [comm_ring R] (S T : set R) :

(ideal.span S) * ideal.span T = ideal.span (⋃ (s : R) (H : s ∈ S) (t : R) (H : t ∈ T), {s * t})

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theorem ideal.span_mul_span' {R : Type u} [comm_ring R] (S T : set R) :

(ideal.span S) * ideal.span T = ideal.span (S * T)

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theorem ideal.span_singleton_mul_span_singleton {R : Type u} [comm_ring R] (r s : R) :

(ideal.span {r}) * ideal.span {s} = ideal.span {r * s}

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theorem ideal.span_singleton_pow {R : Type u} [comm_ring R] (s : R) (n : ℕ) :

ideal.span {s} ^ n = ideal.span {s ^ n}

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theorem ideal.mem_mul_span_singleton {R : Type u} [comm_ring R] {x y : R} {I : ideal R} :

x ∈ I * ideal.span {y} ↔ ∃ (z : R) (H : z ∈ I), z * y = x

source

theorem ideal.mem_span_singleton_mul {R : Type u} [comm_ring R] {x y : R} {I : ideal R} :

x ∈ (ideal.span {y}) * I ↔ ∃ (z : R) (H : z ∈ I), y * z = x

source

theorem ideal.le_span_singleton_mul_iff {R : Type u} [comm_ring R] {x : R} {I J : ideal R} :

I ≤ (ideal.span {x}) * J ↔ ∀ (zI : R), zI ∈ I → (∃ (zJ : R) (H : zJ ∈ J), x * zJ = zI)

source

theorem ideal.span_singleton_mul_le_iff {R : Type u} [comm_ring R] {x : R} {I J : ideal R} :

(ideal.span {x}) * I ≤ J ↔ ∀ (z : R), z ∈ I → x * z ∈ J

source

theorem ideal.span_singleton_mul_le_span_singleton_mul {R : Type u} [comm_ring R] {x y : R} {I J : ideal R} :

(ideal.span {x}) * I ≤ (ideal.span {y}) * J ↔ ∀ (zI : R), zI ∈ I → (∃ (zJ : R) (H : zJ ∈ J), x * zI = y * zJ)

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theorem ideal.eq_span_singleton_mul {R : Type u} [comm_ring R] {x : R} (I J : ideal R) :

I = (ideal.span {x}) * J ↔ (∀ (zI : R), zI ∈ I → (∃ (zJ : R) (H : zJ ∈ J), x * zJ = zI)) ∧ ∀ (z : R), z ∈ J → x * z ∈ I

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theorem ideal.span_singleton_mul_eq_span_singleton_mul {R : Type u} [comm_ring R] {x y : R} (I J : ideal R) :

(ideal.span {x}) * I = (ideal.span {y}) * J ↔ (∀ (zI : R), zI ∈ I → (∃ (zJ : R) (H : zJ ∈ J), x * zI = y * zJ)) ∧ ∀ (zJ : R), zJ ∈ J → (∃ (zI : R) (H : zI ∈ I), x * zI = y * zJ)

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theorem ideal.mul_le_inf {R : Type u} [comm_ring R] {I J : ideal R} :

I * J ≤ I ⊓ J

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theorem ideal.multiset_prod_le_inf {R : Type u} [comm_ring R] {s : multiset (ideal R)} :

s.prod ≤ s.inf

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theorem ideal.prod_le_inf {R : Type u} {ι : Type u_1} [comm_ring R] {s : finset ι} {f : ι → ideal R} :

s.prod f ≤ s.inf f

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theorem ideal.mul_eq_inf_of_coprime {R : Type u} [comm_ring R] {I J : ideal R} (h : I ⊔ J = ⊤) :

I * J = I ⊓ J

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theorem ideal.mul_bot {R : Type u} [comm_ring R] (I : ideal R) :

I * ⊥ = ⊥

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theorem ideal.bot_mul {R : Type u} [comm_ring R] (I : ideal R) :

⊥ * I = ⊥

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theorem ideal.mul_top {R : Type u} [comm_ring R] (I : ideal R) :

I * ⊤ = I

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theorem ideal.top_mul {R : Type u} [comm_ring R] (I : ideal R) :

⊤ * I = I

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theorem ideal.mul_mono {R : Type u} [comm_ring R] {I J K L : ideal R} (hik : I ≤ K) (hjl : J ≤ L) :

I * J ≤ K * L

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theorem ideal.mul_mono_left {R : Type u} [comm_ring R] {I J K : ideal R} (h : I ≤ J) :

I * K ≤ J * K

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theorem ideal.mul_mono_right {R : Type u} [comm_ring R] {I J K : ideal R} (h : J ≤ K) :

I * J ≤ I * K

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theorem ideal.mul_sup {R : Type u} [comm_ring R] (I J K : ideal R) :

I * (J ⊔ K) = I * J ⊔ I * K

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theorem ideal.sup_mul {R : Type u} [comm_ring R] (I J K : ideal R) :

(I ⊔ J) * K = I * K ⊔ J * K

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theorem ideal.pow_le_pow {R : Type u} [comm_ring R] {I : ideal R} {m n : ℕ} (h : m ≤ n) :

I ^ n ≤ I ^ m

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theorem ideal.mul_eq_bot {R : Type u_1} [integral_domain R] {I J : ideal R} :

I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥

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@[instance]

def ideal.no_zero_divisors {R : Type u_1} [integral_domain R] :

no_zero_divisors (ideal R)

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theorem ideal.prod_eq_bot {R : Type u_1} [integral_domain R] {s : multiset (ideal R)} :

s.prod = ⊥ ↔ ∃ (I : ideal R) (H : I ∈ s), I = ⊥

A product of ideals in an integral domain is zero if and only if one of the terms is zero.

source

def ideal.radical {R : Type u} [comm_ring R] (I : ideal R) :

ideal R

The radical of an ideal I consists of the elements r such that r^n ∈ I for some n.

Equations

I.radical = {carrier := {r : R | ∃ (n : ℕ), r ^ n ∈ I}, zero_mem' := _, add_mem' := _, smul_mem' := _}

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theorem ideal.le_radical {R : Type u} [comm_ring R] {I : ideal R} :

I ≤ I.radical

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theorem ideal.radical_top (R : Type u) [comm_ring R] :

⊤.radical = ⊤

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theorem ideal.radical_mono {R : Type u} [comm_ring R] {I J : ideal R} (H : I ≤ J) :

I.radical ≤ J.radical

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@[simp]

theorem ideal.radical_idem {R : Type u} [comm_ring R] (I : ideal R) :

I.radical.radical = I.radical

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theorem ideal.radical_le_radical_iff {R : Type u} [comm_ring R] {I J : ideal R} :

I.radical ≤ J.radical ↔ I ≤ J.radical

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theorem ideal.radical_eq_top {R : Type u} [comm_ring R] {I : ideal R} :

I.radical = ⊤ ↔ I = ⊤

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theorem ideal.is_prime.radical {R : Type u} [comm_ring R] {I : ideal R} (H : I.is_prime) :

I.radical = I

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theorem ideal.radical_sup {R : Type u} [comm_ring R] (I J : ideal R) :

(I ⊔ J).radical = (I.radical ⊔ J.radical).radical

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theorem ideal.radical_inf {R : Type u} [comm_ring R] (I J : ideal R) :

(I ⊓ J).radical = I.radical ⊓ J.radical

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theorem ideal.radical_mul {R : Type u} [comm_ring R] (I J : ideal R) :

(I * J).radical = I.radical ⊓ J.radical

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theorem ideal.is_prime.radical_le_iff {R : Type u} [comm_ring R] {I J : ideal R} (hj : J.is_prime) :

I.radical ≤ J ↔ I ≤ J

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theorem ideal.radical_eq_Inf {R : Type u} [comm_ring R] (I : ideal R) :

I.radical = Inf {J : ideal R | I ≤ J ∧ J.is_prime}

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@[simp]

theorem ideal.radical_bot_of_integral_domain {R : Type u} [integral_domain R] :

⊥.radical = ⊥

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@[instance]

def ideal.comm_semiring {R : Type u} [comm_ring R] :

comm_semiring (ideal R)

Equations

ideal.comm_semiring = submodule.comm_semiring

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theorem ideal.top_pow (R : Type u) [comm_ring R] (n : ℕ) :

⊤ ^ n = ⊤

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theorem ideal.radical_pow {R : Type u} [comm_ring R] (I : ideal R) (n : ℕ) (H : n > 0) :

(I ^ n).radical = I.radical

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theorem ideal.is_prime.mul_le {R : Type u} [comm_ring R] {I J P : ideal R} (hp : P.is_prime) :

I * J ≤ P ↔ I ≤ P ∨ J ≤ P

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theorem ideal.is_prime.inf_le {R : Type u} [comm_ring R] {I J P : ideal R} (hp : P.is_prime) :

I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P

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theorem ideal.is_prime.multiset_prod_le {R : Type u} [comm_ring R] {s : multiset (ideal R)} {P : ideal R} (hp : P.is_prime) (hne : s ≠ 0) :

s.prod ≤ P ↔ ∃ (I : ideal R) (H : I ∈ s), I ≤ P

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theorem ideal.is_prime.multiset_prod_map_le {R : Type u} {ι : Type u_1} [comm_ring R] {s : multiset ι} (f : ι → ideal R) {P : ideal R} (hp : P.is_prime) (hne : s ≠ 0) :

(multiset.map f s).prod ≤ P ↔ ∃ (i : ι) (H : i ∈ s), f i ≤ P

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theorem ideal.is_prime.prod_le {R : Type u} {ι : Type u_1} [comm_ring R] {s : finset ι} {f : ι → ideal R} {P : ideal R} (hp : P.is_prime) (hne : s.nonempty) :

s.prod f ≤ P ↔ ∃ (i : ι) (H : i ∈ s), f i ≤ P

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theorem ideal.is_prime.inf_le' {R : Type u} {ι : Type u_1} [comm_ring R] {s : finset ι} {f : ι → ideal R} {P : ideal R} (hp : P.is_prime) (hsne : s.nonempty) :

s.inf f ≤ P ↔ ∃ (i : ι) (H : i ∈ s), f i ≤ P

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theorem ideal.subset_union {R : Type u} [comm_ring R] {I J K : ideal R} :

↑I ⊆ ↑J ∪ ↑K ↔ I ≤ J ∨ I ≤ K

source

theorem ideal.subset_union_prime' {R : Type u} {ι : Type u_1} [comm_ring R] {s : finset ι} {f : ι → ideal R} {a b : ι} (hp : ∀ (i : ι), i ∈ s → (f i).is_prime) {I : ideal R} :

(↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ (i : ι) (H : i ∈ ↑s), ↑(f i)) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ (i : ι) (H : i ∈ s), I ≤ f i

source

theorem ideal.subset_union_prime {R : Type u} {ι : Type u_1} [comm_ring R] {s : finset ι} {f : ι → ideal R} (a b : ι) (hp : ∀ (i : ι), i ∈ s → i ≠ a → i ≠ b → (f i).is_prime) {I : ideal R} :

(↑I ⊆ ⋃ (i : ι) (H : i ∈ ↑s), ↑(f i)) ↔ ∃ (i : ι) (H : i ∈ s), I ≤ f i

Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6.

source

theorem ideal.le_of_dvd {R : Type u} [comm_ring R] {I J : ideal R} :

I ∣ J → J ≤ I

If I divides J, then I contains J.

In a Dedekind domain, to divide and contain are equivalent, see ideal.dvd_iff_le.

source

theorem ideal.is_unit_iff {R : Type u} [comm_ring R] {I : ideal R} :

is_unit I ↔ I = ⊤

source

@[instance]

def ideal.unique_units {R : Type u} [comm_ring R] :

unique (units (ideal R))

Equations

ideal.unique_units = {to_inhabited := {default := 1}, uniq := _}

source

def ideal.map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (I : ideal R) :

ideal S

I.map f is the span of the image of the ideal I under f, which may be bigger than the image itself.

Equations

ideal.map f I = ideal.span (⇑f '' ↑I)

source

def ideal.comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (I : ideal S) :

ideal R

I.comap f is the preimage of I under f.

Equations

ideal.comap f I = {carrier := ⇑f ⁻¹' ↑I, zero_mem' := _, add_mem' := _, smul_mem' := _}

source

theorem ideal.map_mono {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I J : ideal R} (h : I ≤ J) :

ideal.map f I ≤ ideal.map f J

source

theorem ideal.mem_map_of_mem {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} {x : R} (h : x ∈ I) :

⇑f x ∈ ideal.map f I

source

theorem ideal.apply_coe_mem_map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (I : ideal R) (x : ↥I) :

⇑f ↑x ∈ ideal.map f I

source

theorem ideal.map_le_iff_le_comap {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I : ideal R} {K : ideal S} :

ideal.map f I ≤ K ↔ I ≤ ideal.comap f K

source

@[simp]

theorem ideal.mem_comap {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {K : ideal S} {x : R} :

x ∈ ideal.comap f K ↔ ⇑f x ∈ K

source

theorem ideal.comap_mono {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {K L : ideal S} (h : K ≤ L) :

ideal.comap f K ≤ ideal.comap f L

source

theorem ideal.comap_ne_top {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {K : ideal S} (hK : K ≠ ⊤) :

ideal.comap f K ≠ ⊤

source

@[instance]

def ideal.is_prime.comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {K : ideal S} [hK : K.is_prime] :

(ideal.comap f K).is_prime

source

theorem ideal.map_top {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

ideal.map f ⊤ = ⊤

source

theorem ideal.gc_map_comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

galois_connection (ideal.map f) (ideal.comap f)

source

@[simp]

theorem ideal.comap_id {R : Type u} [ring R] (I : ideal R) :

ideal.comap (ring_hom.id R) I = I

source

@[simp]

theorem ideal.map_id {R : Type u} [ring R] (I : ideal R) :

ideal.map (ring_hom.id R) I = I

source

theorem ideal.comap_comap {R : Type u} {S : Type v} [ring R] [ring S] {T : Type u_1} [ring T] {I : ideal T} (f : R →+* S) (g : S →+* T) :

ideal.comap f (ideal.comap g I) = ideal.comap (g.comp f) I

source

theorem ideal.map_map {R : Type u} {S : Type v} [ring R] [ring S] {T : Type u_1} [ring T] {I : ideal R} (f : R →+* S) (g : S →+* T) :

ideal.map g (ideal.map f I) = ideal.map (g.comp f) I

source

theorem ideal.map_span {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set R) :

ideal.map f (ideal.span s) = ideal.span (⇑f '' s)

source

theorem ideal.map_le_of_le_comap {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I : ideal R} {K : ideal S} :

I ≤ ideal.comap f K → ideal.map f I ≤ K

source

theorem ideal.le_comap_of_map_le {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I : ideal R} {K : ideal S} :

ideal.map f I ≤ K → I ≤ ideal.comap f K

source

theorem ideal.le_comap_map {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I : ideal R} :

I ≤ ideal.comap f (ideal.map f I)

source

theorem ideal.map_comap_le {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {K : ideal S} :

ideal.map f (ideal.comap f K) ≤ K

source

@[simp]

theorem ideal.comap_top {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} :

ideal.comap f ⊤ = ⊤

source

@[simp]

theorem ideal.comap_eq_top_iff {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} {I : ideal S} :

ideal.comap f I = ⊤ ↔ I = ⊤

source

@[simp]

theorem ideal.map_bot {R : Type u} {S : Type v} [ring R] [ring S] {f : R →+* S} :

ideal.map f ⊥ = ⊥

source

@[simp]

theorem ideal.map_comap_map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (I : ideal R) :

ideal.map f (ideal.comap f (ideal.map f I)) = ideal.map f I

source

@[simp]

theorem ideal.comap_map_comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (K : ideal S) :

ideal.comap f (ideal.map f (ideal.comap f K)) = ideal.comap f K

source

theorem ideal.map_sup {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (I J : ideal R) :

ideal.map f (I ⊔ J) = ideal.map f I ⊔ ideal.map f J

source

theorem ideal.comap_inf {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (K L : ideal S) :

ideal.comap f (K ⊓ L) = ideal.comap f K ⊓ ideal.comap f L

source

theorem ideal.map_supr {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {ι : Sort u_1} (K : ι → ideal R) :

ideal.map f (supr K) = ⨆ (i : ι), ideal.map f (K i)

source

theorem ideal.comap_infi {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {ι : Sort u_1} (K : ι → ideal S) :

ideal.comap f (infi K) = ⨅ (i : ι), ideal.comap f (K i)

source

theorem ideal.map_Sup {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set (ideal R)) :

ideal.map f (Sup s) = ⨆ (I : ideal R) (H : I ∈ s), ideal.map f I

source

theorem ideal.comap_Inf {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set (ideal S)) :

ideal.comap f (Inf s) = ⨅ (I : ideal S) (H : I ∈ s), ideal.comap f I

source

theorem ideal.comap_Inf' {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (s : set (ideal S)) :

ideal.comap f (Inf s) = ⨅ (I : ideal R) (H : I ∈ ideal.comap f '' s), I

source

theorem ideal.comap_is_prime {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (K : ideal S) [H : K.is_prime] :

(ideal.comap f K).is_prime

source

theorem ideal.map_inf_le {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I J : ideal R} :

ideal.map f (I ⊓ J) ≤ ideal.map f I ⊓ ideal.map f J

source

theorem ideal.le_comap_sup {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {K L : ideal S} :

ideal.comap f K ⊔ ideal.comap f L ≤ ideal.comap f (K ⊔ L)

source

theorem ideal.map_comap_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) (I : ideal S) :

ideal.map f (ideal.comap f I) = I

source

def ideal.gi_map_comap {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

galois_insertion (ideal.map f) (ideal.comap f)

map and comap are adjoint, and the composition map f ∘ comap f is the identity

Equations

ideal.gi_map_comap f hf = galois_insertion.monotone_intro _ _ _ _

source

theorem ideal.map_surjective_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

function.surjective (ideal.map f)

source

theorem ideal.comap_injective_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

function.injective (ideal.comap f)

source

theorem ideal.map_sup_comap_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) (I J : ideal S) :

ideal.map f (ideal.comap f I ⊔ ideal.comap f J) = I ⊔ J

source

theorem ideal.map_supr_comap_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {ι : Sort u_1} (hf : function.surjective ⇑f) (K : ι → ideal S) :

ideal.map f (⨆ (i : ι), ideal.comap f (K i)) = supr K

source

theorem ideal.map_inf_comap_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) (I J : ideal S) :

ideal.map f (ideal.comap f I ⊓ ideal.comap f J) = I ⊓ J

source

theorem ideal.map_infi_comap_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {ι : Sort u_1} (hf : function.surjective ⇑f) (K : ι → ideal S) :

ideal.map f (⨅ (i : ι), ideal.comap f (K i)) = infi K

source

theorem ideal.mem_image_of_mem_map_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) {I : ideal R} {y : S} (H : y ∈ ideal.map f I) :

y ∈ ⇑f '' ↑I

source

theorem ideal.mem_map_iff_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) {I : ideal R} {y : S} :

y ∈ ideal.map f I ↔ ∃ (x : R), x ∈ I ∧ ⇑f x = y

source

theorem ideal.comap_map_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) (I : ideal R) :

ideal.comap f (ideal.map f I) = I ⊔ ideal.comap f ⊥

source

theorem ideal.le_map_of_comap_le_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} {K : ideal S} (hf : function.surjective ⇑f) :

ideal.comap f K ≤ I → K ≤ ideal.map f I

source

def ideal.rel_iso_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

ideal S ≃o {p // ideal.comap f ⊥ ≤ p}

Correspondence theorem

Equations

ideal.rel_iso_of_surjective f hf = {to_equiv := {to_fun := λ (J : ideal S), ⟨ideal.comap f J, _⟩, inv_fun := λ (I : {p // ideal.comap f ⊥ ≤ p}), ideal.map f I.val, left_inv := _, right_inv := _}, map_rel_iff' := _}

source

def ideal.order_embedding_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.surjective ⇑f) :

ideal S ↪o ideal R

The map on ideals induced by a surjective map preserves inclusion.

Equations

ideal.order_embedding_of_surjective f hf = (rel_iso.to_rel_embedding (ideal.rel_iso_of_surjective f hf)).trans (subtype.rel_embedding has_le.le (λ (p : ideal R), ideal.comap f ⊥ ≤ p))

source

theorem ideal.map_eq_top_or_is_maximal_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} (hf : function.surjective ⇑f) (H : I.is_maximal) :

ideal.map f I = ⊤ ∨ (ideal.map f I).is_maximal

source

theorem ideal.comap_is_maximal_of_surjective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {K : ideal S} (hf : function.surjective ⇑f) [H : K.is_maximal] :

(ideal.comap f K).is_maximal

source

@[simp]

theorem ideal.map_of_equiv {R : Type u} {S : Type v} [ring R] [ring S] (I : ideal R) (f : R ≃+* S) :

ideal.map ↑(f.symm) (ideal.map ↑f I) = I

If f : R ≃+* S is a ring isomorphism and I : ideal R, then map f (map f.symm) = I.

source

@[simp]

theorem ideal.comap_of_equiv {R : Type u} {S : Type v} [ring R] [ring S] (I : ideal R) (f : R ≃+* S) :

ideal.comap ↑f (ideal.comap ↑(f.symm) I) = I

If f : R ≃+* S is a ring isomorphism and I : ideal R, then comap f.symm (comap f) = I.

source

theorem ideal.map_comap_of_equiv {R : Type u} {S : Type v} [ring R] [ring S] (I : ideal R) (f : R ≃+* S) :

ideal.map ↑f I = ideal.comap ↑(f.symm) I

If f : R ≃+* S is a ring isomorphism and I : ideal R, then map f I = comap f.symm I.

source

theorem ideal.comap_bot_le_of_injective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} (hf : function.injective ⇑f) :

ideal.comap f ⊥ ≤ I

source

def ideal.rel_iso_of_bijective {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) (hf : function.bijective ⇑f) :

ideal S ≃o ideal R

Special case of the correspondence theorem for isomorphic rings

Equations

ideal.rel_iso_of_bijective f hf = {to_equiv := {to_fun := ideal.comap f, inv_fun := ideal.map f, left_inv := _, right_inv := _}, map_rel_iff' := _}

source

theorem ideal.comap_le_iff_le_map {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} {K : ideal S} (hf : function.bijective ⇑f) :

ideal.comap f K ≤ I ↔ K ≤ ideal.map f I

source

theorem ideal.map.is_maximal {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {I : ideal R} (hf : function.bijective ⇑f) (H : I.is_maximal) :

(ideal.map f I).is_maximal

source

theorem ideal.ring_equiv.bot_maximal_iff {R : Type u} {S : Type v} [ring R] [ring S] (e : R ≃+* S) :

⊥.is_maximal ↔ ⊥.is_maximal

source

theorem ideal.mem_quotient_iff_mem {R : Type u} [comm_ring R] {I J : ideal R} (hIJ : I ≤ J) {x : R} :

⇑(ideal.quotient.mk I) x ∈ ideal.map (ideal.quotient.mk I) J ↔ x ∈ J

source

theorem ideal.map_mul {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) (I J : ideal R) :

ideal.map f (I * J) = (ideal.map f I) * ideal.map f J

source

theorem ideal.comap_radical {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) (K : ideal S) :

ideal.comap f K.radical = (ideal.comap f K).radical

source

@[simp]

theorem ideal.map_quotient_self {R : Type u} [comm_ring R] (I : ideal R) :

ideal.map (ideal.quotient.mk I) I = ⊥

source

theorem ideal.map_radical_le {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) {I : ideal R} :

ideal.map f I.radical ≤ (ideal.map f I).radical

source

theorem ideal.le_comap_mul {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) {K L : ideal S} :

(ideal.comap f K) * ideal.comap f L ≤ ideal.comap f (K * L)

source

def ideal.is_primary {R : Type u} [comm_ring R] (I : ideal R) :

Prop

A proper ideal I is primary iff xy ∈ I implies x ∈ I or y ∈ radical I.

Equations

I.is_primary = (I ≠ ⊤ ∧ ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ I.radical)

source

theorem ideal.is_primary.to_is_prime {R : Type u} [comm_ring R] (I : ideal R) (hi : I.is_prime) :

I.is_primary

source

theorem ideal.mem_radical_of_pow_mem {R : Type u} [comm_ring R] {I : ideal R} {x : R} {m : ℕ} (hx : x ^ m ∈ I.radical) :

x ∈ I.radical

source

theorem ideal.is_prime_radical {R : Type u} [comm_ring R] {I : ideal R} (hi : I.is_primary) :

I.radical.is_prime

source

theorem ideal.is_primary_inf {R : Type u} [comm_ring R] {I J : ideal R} (hi : I.is_primary) (hj : J.is_primary) (hij : I.radical = J.radical) :

(I ⊓ J).is_primary

source

def ring_hom.ker {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

ideal R

Kernel of a ring homomorphism as an ideal of the domain.

Equations

f.ker = ideal.comap f ⊥

source

theorem ring_hom.mem_ker {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) {r : R} :

r ∈ f.ker ↔ ⇑f r = 0

An element is in the kernel if and only if it maps to zero.

source

theorem ring_hom.ker_eq {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

↑(f.ker) = ⇑f ⁻¹' {0}

source

theorem ring_hom.ker_eq_comap_bot {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

f.ker = ideal.comap f ⊥

source

theorem ring_hom.injective_iff_ker_eq_bot {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

function.injective ⇑f ↔ f.ker = ⊥

source

theorem ring_hom.ker_eq_bot_iff_eq_zero {R : Type u} {S : Type v} [ring R] [ring S] (f : R →+* S) :

f.ker = ⊥ ↔ ∀ (x : R), ⇑f x = 0 → x = 0

source

theorem ring_hom.not_one_mem_ker {R : Type u} {S : Type v} [ring R] [ring S] [nontrivial S] (f : R →+* S) :

1 ∉ f.ker

If the target is not the zero ring, then one is not in the kernel.

source

@[simp]

theorem ring_hom.ker_coe_equiv {R : Type u} {S : Type v} [ring R] [ring S] (f : R ≃+* S) :

↑f.ker = ⊥

source

def ring_hom.ker_lift {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) :

f.ker.quotient →+* S

The induced map from the quotient by the kernel to the codomain.

This is an isomorphism if f has a right inverse (quotient_ker_equiv_of_right_inverse) / is surjective (quotient_ker_equiv_of_surjective).

Equations

f.ker_lift = ideal.quotient.lift f.ker f _

source

@[simp]

theorem ring_hom.ker_lift_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) (r : R) :

⇑(f.ker_lift) (⇑(ideal.quotient.mk f.ker) r) = ⇑f r

source

theorem ring_hom.ker_lift_injective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) :

function.injective ⇑(f.ker_lift)

The induced map from the quotient by the kernel is injective.

source

def ring_hom.quotient_ker_equiv_of_right_inverse {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} {g : S → R} (hf : function.right_inverse g ⇑f) :

f.ker.quotient ≃+* S

The first isomorphism theorem for commutative rings, computable version.

Equations

ring_hom.quotient_ker_equiv_of_right_inverse hf = {to_fun := ⇑(f.ker_lift), inv_fun := ⇑(ideal.quotient.mk f.ker) ∘ g, left_inv := _, right_inv := hf, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_hom.quotient_ker_equiv_of_right_inverse.apply {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} {g : S → R} (hf : function.right_inverse g ⇑f) (x : f.ker.quotient) :

⇑(ring_hom.quotient_ker_equiv_of_right_inverse hf) x = ⇑(f.ker_lift) x

source

@[simp]

theorem ring_hom.quotient_ker_equiv_of_right_inverse.symm.apply {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} {g : S → R} (hf : function.right_inverse g ⇑f) (x : S) :

⇑((ring_hom.quotient_ker_equiv_of_right_inverse hf).symm) x = ⇑(ideal.quotient.mk f.ker) (g x)

source

def ring_hom.quotient_ker_equiv_of_surjective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} (hf : function.surjective ⇑f) :

f.ker.quotient ≃+* S

The first isomorphism theorem for commutative rings.

Equations

ring_hom.quotient_ker_equiv_of_surjective hf = ring_hom.quotient_ker_equiv_of_right_inverse _

source

theorem ring_hom.ker_is_prime {R : Type u} {S : Type v} [ring R] [integral_domain S] (f : R →+* S) :

f.ker.is_prime

The kernel of a homomorphism to an integral domain is a prime ideal.

source

theorem ring_hom.ker_is_maximal_of_surjective {R : Type u_1} {K : Type u_2} [ring R] [field K] (f : R →+* K) (hf : function.surjective ⇑f) :

f.ker.is_maximal

The kernel of a homomorphism to a field is a maximal ideal.

source

theorem ideal.map_eq_bot_iff_le_ker {R : Type u_1} {S : Type u_2} [ring R] [ring S] {I : ideal R} (f : R →+* S) :

ideal.map f I = ⊥ ↔ I ≤ f.ker

source

theorem ideal.ker_le_comap {R : Type u_1} {S : Type u_2} [ring R] [ring S] {K : ideal S} (f : R →+* S) :

f.ker ≤ ideal.comap f K

source

theorem ideal.map_Inf {R : Type u_1} {S : Type u_2} [ring R] [ring S] {A : set (ideal R)} {f : R →+* S} (hf : function.surjective ⇑f) :

(∀ (J : ideal R), J ∈ A → f.ker ≤ J) → ideal.map f (Inf A) = Inf (ideal.map f '' A)

source

theorem ideal.map_is_prime_of_surjective {R : Type u_1} {S : Type u_2} [ring R] [ring S] {f : R →+* S} (hf : function.surjective ⇑f) {I : ideal R} [H : I.is_prime] (hk : f.ker ≤ I) :

(ideal.map f I).is_prime

source

theorem ideal.map_is_prime_of_equiv {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R ≃+* S) {I : ideal R} [I.is_prime] :

(ideal.map ↑f I).is_prime

source

@[simp]

theorem ideal.mk_ker {R : Type u_1} [comm_ring R] {I : ideal R} :

(ideal.quotient.mk I).ker = I

source

theorem ideal.map_mk_eq_bot_of_le {R : Type u_1} [comm_ring R] {I J : ideal R} (h : I ≤ J) :

ideal.map (ideal.quotient.mk J) I = ⊥

source

theorem ideal.ker_quotient_lift {R : Type u_1} [comm_ring R] {S : Type v} [comm_ring S] {I : ideal R} (f : R →+* S) (H : I ≤ f.ker) :

(ideal.quotient.lift I f H).ker = ideal.map (ideal.quotient.mk I) f.ker

source

theorem ideal.map_eq_iff_sup_ker_eq_of_surjective {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {I J : ideal R} (f : R →+* S) (hf : function.surjective ⇑f) :

ideal.map f I = ideal.map f J ↔ I ⊔ f.ker = J ⊔ f.ker

source

theorem ideal.map_radical_of_surjective {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {f : R →+* S} (hf : function.surjective ⇑f) {I : ideal R} (h : f.ker ≤ I) :

ideal.map f I.radical = (ideal.map f I).radical

source

@[simp]

theorem ideal.bot_quotient_is_maximal_iff {R : Type u_1} [comm_ring R] (I : ideal R) :

⊥.is_maximal ↔ I.is_maximal

source

@[instance]

def ideal.quotient.algebra (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {I : ideal A} :

algebra R I.quotient

The R-algebra structure on A/I for an R-algebra A

Equations

ideal.quotient.algebra R = ((ideal.quotient.mk I).comp (algebra_map R A)).to_algebra

source

def ideal.quotient.mkₐ (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

A →ₐ[R] I.quotient

The canonical morphism A →ₐ[R] I.quotient as morphism of R-algebras, for I an ideal of A, where A is an R-algebra.

Equations

ideal.quotient.mkₐ R I = {to_fun := λ (a : A), submodule.quotient.mk a, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _, commutes' := _}

source

theorem ideal.quotient.alg_map_eq (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

algebra_map R I.quotient = (algebra_map A I.quotient).comp (algebra_map R A)

source

@[instance]

def ideal.quotient.is_scalar_tower (R : Type u_1) {S : Type u_2} [comm_ring R] [comm_ring S] {A : Type u_3} [comm_ring A] [algebra R A] [algebra S A] [algebra S R] [is_scalar_tower S R A] {I : ideal A} :

is_scalar_tower S R I.quotient

source

theorem ideal.quotient.mkₐ_to_ring_hom (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

(ideal.quotient.mkₐ R I).to_ring_hom = ideal.quotient.mk I

source

@[simp]

theorem ideal.quotient.mkₐ_eq_mk (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

⇑(ideal.quotient.mkₐ R I) = ⇑(ideal.quotient.mk I)

source

theorem ideal.quotient.mkₐ_surjective (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

function.surjective ⇑(ideal.quotient.mkₐ R I)

The canonical morphism A →ₐ[R] I.quotient is surjective.

source

@[simp]

theorem ideal.quotient.mkₐ_ker (R : Type u_1) [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] (I : ideal A) :

↑(ideal.quotient.mkₐ R I).ker = I

The kernel of A →ₐ[R] I.quotient is I.

source

theorem ideal.ker_lift.map_smul {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] (f : A →ₐ[R] B) (r : R) (x : f.to_ring_hom.ker.quotient) :

⇑(f.to_ring_hom.ker_lift) (r • x) = r • ⇑(f.to_ring_hom.ker_lift) x

source

def ideal.ker_lift_alg {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] (f : A →ₐ[R] B) :

f.to_ring_hom.ker.quotient →ₐ[R] B

The induced algebras morphism from the quotient by the kernel to the codomain.

This is an isomorphism if f has a right inverse (quotient_ker_alg_equiv_of_right_inverse) / is surjective (quotient_ker_alg_equiv_of_surjective).

Equations

ideal.ker_lift_alg f = alg_hom.mk' f.to_ring_hom.ker_lift _

source

@[simp]

theorem ideal.ker_lift_alg_mk {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] (f : A →ₐ[R] B) (a : A) :

⇑(ideal.ker_lift_alg f) (⇑(ideal.quotient.mk f.to_ring_hom.ker) a) = ⇑f a

source

@[simp]

theorem ideal.ker_lift_alg_to_ring_hom {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] (f : A →ₐ[R] B) :

(ideal.ker_lift_alg f).to_ring_hom = ↑f.ker_lift

source

theorem ideal.ker_lift_alg_injective {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] (f : A →ₐ[R] B) :

function.injective ⇑(ideal.ker_lift_alg f)

The induced algebra morphism from the quotient by the kernel is injective.

source

def ideal.quotient_ker_alg_equiv_of_right_inverse {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] {f : A →ₐ[R] B} {g : B → A} (hf : function.right_inverse g ⇑f) :

f.to_ring_hom.ker.quotient ≃ₐ[R] B

The first isomorphism theorem for agebras, computable version.

Equations

ideal.quotient_ker_alg_equiv_of_right_inverse hf = {to_fun := (ring_hom.quotient_ker_equiv_of_right_inverse _).to_fun, inv_fun := (ring_hom.quotient_ker_equiv_of_right_inverse _).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _, commutes' := _}

source

@[simp]

theorem ideal.quotient_ker_alg_equiv_of_right_inverse.apply {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] {f : A →ₐ[R] B} {g : B → A} (hf : function.right_inverse g ⇑f) (x : f.to_ring_hom.ker.quotient) :

⇑(ideal.quotient_ker_alg_equiv_of_right_inverse hf) x = ⇑(ideal.ker_lift_alg f) x

source

@[simp]

theorem ideal.quotient_ker_alg_equiv_of_right_inverse_symm.apply {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] {f : A →ₐ[R] B} {g : B → A} (hf : function.right_inverse g ⇑f) (x : B) :

⇑((ideal.quotient_ker_alg_equiv_of_right_inverse hf).symm) x = ⇑(ideal.quotient.mkₐ R f.to_ring_hom.ker) (g x)

source

def ideal.quotient_ker_alg_equiv_of_surjective {R : Type u_1} [comm_ring R] {A : Type u_3} [comm_ring A] [algebra R A] {B : Type u_4} [comm_ring B] [algebra R B] {f : A →ₐ[R] B} (hf : function.surjective ⇑f) :

f.to_ring_hom.ker.quotient ≃ₐ[R] B

The first isomorphism theorem for algebras.

Equations

ideal.quotient_ker_alg_equiv_of_surjective hf = ideal.quotient_ker_alg_equiv_of_right_inverse _

source

def ideal.quotient_map {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {I : ideal R} (J : ideal S) (f : R →+* S) (hIJ : I ≤ ideal.comap f J) :

I.quotient →+* J.quotient

The ring hom R/I →+* S/J induced by a ring hom f : R →+* S with I ≤ f⁻¹(J)

Equations

J.quotient_map f hIJ = ideal.quotient.lift I ((ideal.quotient.mk J).comp f) _

source

@[simp]

theorem ideal.quotient_map_mk {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ ideal.comap f I} {x : R} :

⇑(I.quotient_map f H) (⇑(ideal.quotient.mk J) x) = ⇑(ideal.quotient.mk I) (⇑f x)

source

theorem ideal.quotient_map_comp_mk {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} (H : J ≤ ideal.comap f I) :

(I.quotient_map f H).comp (ideal.quotient.mk J) = (ideal.quotient.mk I).comp f

source

@[simp]

theorem ideal.quotient_equiv_symm_apply {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map ↑f I) (ᾰ : J.quotient) :

⇑((I.quotient_equiv J f hIJ).symm) ᾰ = ⇑(I.quotient_map ↑(f.symm) _) ᾰ

source

def ideal.quotient_equiv {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map ↑f I) :

I.quotient ≃+* J.quotient

The ring equiv R/I ≃+* S/J induced by a ring equiv f : R ≃+** S, where J = f(I).

Equations

I.quotient_equiv J f hIJ = {to_fun := (J.quotient_map ↑f _).to_fun, inv_fun := ⇑(I.quotient_map ↑(f.symm) _), left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ideal.quotient_equiv_apply {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map ↑f I) (ᾰ : I.quotient) :

⇑(I.quotient_equiv J f hIJ) ᾰ = (J.quotient_map ↑f _).to_fun ᾰ

source

theorem ideal.quotient_map_injective' {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ ideal.comap f I} (h : ideal.comap f I ≤ J) :

function.injective ⇑(I.quotient_map f H)

H and h are kept as separate hypothesis since H is used in constructing the quotient map.

source

theorem ideal.quotient_map_injective {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {I : ideal S} {f : R →+* S} :

function.injective ⇑(I.quotient_map f le_rfl)

If we take J = I.comap f then quotient_map is injective automatically.

source

theorem ideal.quotient_map_surjective {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ ideal.comap f I} (hf : function.surjective ⇑f) :

function.surjective ⇑(I.quotient_map f H)

source

theorem ideal.comp_quotient_map_eq_of_comp_eq {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {R' : Type u_3} {S' : Type u_4} [comm_ring R'] [comm_ring S'] {f : R →+* S} {f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f) (I : ideal S') :

(I.quotient_map g' le_rfl).comp ((ideal.comap g' I).quotient_map f le_rfl) = (I.quotient_map f' le_rfl).comp ((ideal.comap f' I).quotient_map g _)

Commutativity of a square is preserved when taking quotients by an ideal.

source

def ideal.quotient_mapₐ {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {A : Type u_3} [comm_ring A] [algebra R A] [algebra R S] {I : ideal A} (J : ideal S) (f : A →ₐ[R] S) (hIJ : I ≤ ideal.comap ↑f J) :

I.quotient →ₐ[R] J.quotient

The algebra hom A/I →+* S/J induced by an algebra hom f : A →ₐ[R] S with I ≤ f⁻¹(J).

Equations

J.quotient_mapₐ f hIJ = {to_fun := (J.quotient_map ↑f hIJ).to_fun, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _, commutes' := _}

source

@[simp]

theorem ideal.quotient_map_mkₐ {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {A : Type u_3} [comm_ring A] [algebra R A] [algebra R S] {I : ideal A} (J : ideal S) (f : A →ₐ[R] S) (H : I ≤ ideal.comap ↑f J) {x : A} :

⇑(J.quotient_mapₐ f H) (⇑(ideal.quotient.mk I) x) = ⇑(ideal.quotient.mkₐ R J) (⇑f x)

source

theorem ideal.quotient_map_comp_mkₐ {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {A : Type u_3} [comm_ring A] [algebra R A] [algebra R S] {I : ideal A} (J : ideal S) (f : A →ₐ[R] S) (H : I ≤ ideal.comap ↑f J) :

(J.quotient_mapₐ f H).comp (ideal.quotient.mkₐ R I) = (ideal.quotient.mkₐ R J).comp f

source

def ideal.quotient_equiv_alg {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {A : Type u_3} [comm_ring A] [algebra R A] [algebra R S] (I : ideal A) (J : ideal S) (f : A ≃ₐ[R] S) (hIJ : J = ideal.map ↑f I) :

I.quotient ≃ₐ[R] J.quotient

The algebra equiv A/I ≃ₐ[R] S/J induced by an algebra equiv f : A ≃ₐ[R] S, whereJ = f(I).

Equations

I.quotient_equiv_alg J f hIJ = {to_fun := (I.quotient_equiv J ↑f hIJ).to_fun, inv_fun := (I.quotient_equiv J ↑f hIJ).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _, commutes' := _}

source

@[instance]

def ideal.quotient_algebra {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal S} [algebra R S] :

algebra (ideal.comap (algebra_map R S) J).quotient J.quotient

Equations

ideal.quotient_algebra = (J.quotient_map (algebra_map R S) ideal.quotient_algebra._proof_1).to_algebra

source

theorem ideal.algebra_map_quotient_injective {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {J : ideal S} [algebra R S] :

function.injective ⇑(algebra_map (ideal.comap (algebra_map R S) J).quotient J.quotient)

source

@[instance]

def submodule.module_submodule {R : Type u} {M : Type v} [comm_ring R] [add_comm_group M] [module R M] :

module (ideal R) (submodule R M)

Equations

submodule.module_submodule = {to_distrib_mul_action := {to_mul_action := {to_has_scalar := submodule.has_scalar' _inst_3, one_smul := _, mul_smul := _}, smul_add := _, smul_zero := _}, add_smul := _, zero_smul := _}

source

def ring_hom.lift_of_right_inverse_aux {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) (g : A →+* C) (hg : f.ker ≤ g.ker) :

B →+* C

Auxiliary definition used to define lift_of_right_inverse

Equations

f.lift_of_right_inverse_aux f_inv hf g hg = {to_fun := λ (b : B), ⇑g (f_inv b), map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

source

@[simp]

theorem ring_hom.lift_of_right_inverse_aux_comp_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) (g : A →+* C) (hg : f.ker ≤ g.ker) (a : A) :

⇑(f.lift_of_right_inverse_aux f_inv hf g hg) (⇑f a) = ⇑g a

source

def ring_hom.lift_of_right_inverse {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) :

{g // f.ker ≤ g.ker} ≃ (B →+* C)

lift_of_right_inverse f hf g hg is the unique ring homomorphism φ

such that φ.comp f = g (ring_hom.lift_of_right_inverse_comp),
where f : A →+* B is has a right_inverse f_inv (hf),
and g : B →+* C satisfies hg : f.ker ≤ g.ker.

See ring_hom.eq_lift_of_right_inverse for the uniqueness lemma.

   A .
   |  \
 f |   \ g
   |    \
   v     \⌟
   B ----> C
      ∃!φ

Equations

f.lift_of_right_inverse f_inv hf = {to_fun := λ (g : {g // f.ker ≤ g.ker}), f.lift_of_right_inverse_aux f_inv hf g.val _, inv_fun := λ (φ : B →+* C), ⟨φ.comp f, _⟩, left_inv := _, right_inv := _}

source

@[simp]

def ring_hom.lift_of_surjective {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (hf : function.surjective ⇑f) :

{g // f.ker ≤ g.ker} ≃ (B →+* C)

A non-computable version of ring_hom.lift_of_right_inverse for when no computable right inverse is available, that uses function.surj_inv.

source

theorem ring_hom.lift_of_right_inverse_comp_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) (g : {g // f.ker ≤ g.ker}) (x : A) :

⇑(⇑(f.lift_of_right_inverse f_inv hf) g) (⇑f x) = ⇑g x

source

theorem ring_hom.lift_of_right_inverse_comp {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) (g : {g // f.ker ≤ g.ker}) :

(⇑(f.lift_of_right_inverse f_inv hf) g).comp f = ↑g

source

theorem ring_hom.eq_lift_of_right_inverse {A : Type u_1} {B : Type u_2} {C : Type u_3} [ring A] [ring B] [ring C] (f : A →+* B) (f_inv : B → A) (hf : function.right_inverse f_inv ⇑f) (g : A →+* C) (hg : f.ker ≤ g.ker) (h : B →+* C) (hh : h.comp f = g) :

h = ⇑(f.lift_of_right_inverse f_inv hf) ⟨g, hg⟩

source

def double_quot.quot_left_to_quot_sup {R : Type u} [comm_ring R] (I J : ideal R) :

I.quotient →+* (I ⊔ J).quotient

The obvious ring hom R/I → R/(I ⊔ J)

Equations

double_quot.quot_left_to_quot_sup I J = ideal.quotient.factor I (I ⊔ J) _

source

theorem double_quot.ker_quot_left_to_quot_sup {R : Type u} [comm_ring R] (I J : ideal R) :

(double_quot.quot_left_to_quot_sup I J).ker = ideal.map (ideal.quotient.mk I) J

The kernel of quot_left_to_quot_sup

source

def double_quot.quot_quot_to_quot_sup {R : Type u} [comm_ring R] (I J : ideal R) :

(ideal.map (ideal.quotient.mk I) J).quotient →+* (I ⊔ J).quotient

The ring homomorphism (R/I)/J' -> R/(I ⊔ J) induced by quot_left_to_quot_sup where J' is the image of J in R/I

Equations

double_quot.quot_quot_to_quot_sup I J = ideal.quotient.lift (ideal.map (ideal.quotient.mk I) J) (double_quot.quot_left_to_quot_sup I J) _

source

def double_quot.quot_quot_mk {R : Type u} [comm_ring R] (I J : ideal R) :

R →+* (ideal.map (ideal.quotient.mk I) J).quotient

The composite of the maps R → (R/I) and (R/I) → (R/I)/J'

Equations

double_quot.quot_quot_mk I J = (ideal.quotient.mk (ideal.map (ideal.quotient.mk I) J)).comp (ideal.quotient.mk I)

source

theorem double_quot.ker_quot_quot_mk {R : Type u} [comm_ring R] (I J : ideal R) :

(double_quot.quot_quot_mk I J).ker = I ⊔ J

The kernel of quot_quot_mk

source

def double_quot.lift_sup_quot_quot_mk {R : Type u} [comm_ring R] (I J : ideal R) :

(I ⊔ J).quotient →+* (ideal.map (ideal.quotient.mk I) J).quotient

The ring homomorphism R/(I ⊔ J) → (R/I)/J'induced by quot_quot_mk

Equations

double_quot.lift_sup_quot_quot_mk I J = ideal.quotient.lift (I ⊔ J) (double_quot.quot_quot_mk I J) _

source

def double_quot.quot_quot_equiv_quot_sup {R : Type u} [comm_ring R] (I J : ideal R) :

(ideal.map (ideal.quotient.mk I) J).quotient ≃+* (I ⊔ J).quotient

quot_quot_to_quot_add and lift_sup_double_qot_mk are inverse isomorphisms

Equations

double_quot.quot_quot_equiv_quot_sup I J = ring_equiv.of_hom_inv (double_quot.quot_quot_to_quot_sup I J) (double_quot.lift_sup_quot_quot_mk I J) _ _

mathlib documentation

More operations on modules and ideals #