Ideals over a ring #

This file defines Ideal R, the type of (left) ideals over a ring R. Note that over commutative rings, left ideals and two-sided ideals are equivalent.

Implementation notes #

Ideal R is implemented using Submodule R R, where • is interpreted as *.

TODO #

Support right ideals, and two-sided ideals over non-commutative rings.

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@[reducible]

def Ideal (R : Type u) [Semiring R] :

Type u

A (left) ideal in a semiring R is an additive submonoid s such that a * b ∈ s whenever b ∈ s. If R is a ring, then s is an additive subgroup.

Equations

Ideal R = Submodule R R

Instances For

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theorem isPrincipalIdealRing_iff (R : Type u) [Semiring R] :

IsPrincipalIdealRing R ↔ ∀ (S : Ideal R), Submodule.IsPrincipal S

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class IsPrincipalIdealRing (R : Type u) [Semiring R] :

Prop

A ring is a principal ideal ring if all (left) ideals are principal.

principal : ∀ (S : Ideal R), Submodule.IsPrincipal S

Instances

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theorem Ideal.zero_mem {α : Type u} [Semiring α] (I : Ideal α) :

0 ∈ I

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theorem Ideal.add_mem {α : Type u} [Semiring α] (I : Ideal α) {a : α} {b : α} :

a ∈ I → b ∈ I → a + b ∈ I

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theorem Ideal.mul_mem_left {α : Type u} [Semiring α] (I : Ideal α) (a : α) {b : α} :

b ∈ I → a * b ∈ I

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theorem Ideal.ext {α : Type u} [Semiring α] {I : Ideal α} {J : Ideal α} (h : ∀ (x : α), x ∈ I ↔ x ∈ J) :

I = J

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theorem Ideal.sum_mem {α : Type u} [Semiring α] (I : Ideal α) {ι : Type u_1} {t : Finset ι} {f : ι → α} :

(∀ c ∈ t, f c ∈ I) → (Finset.sum t fun (i : ι) => f i) ∈ I

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theorem Ideal.eq_top_of_unit_mem {α : Type u} [Semiring α] (I : Ideal α) (x : α) (y : α) (hx : x ∈ I) (h : y * x = 1) :

I = ⊤

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theorem Ideal.eq_top_of_isUnit_mem {α : Type u} [Semiring α] (I : Ideal α) {x : α} (hx : x ∈ I) (h : IsUnit x) :

I = ⊤

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theorem Ideal.eq_top_iff_one {α : Type u} [Semiring α] (I : Ideal α) :

I = ⊤ ↔ 1 ∈ I

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theorem Ideal.ne_top_iff_one {α : Type u} [Semiring α] (I : Ideal α) :

I ≠ ⊤ ↔ 1 ∉ I

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@[simp]

theorem Ideal.unit_mul_mem_iff_mem {α : Type u} [Semiring α] (I : Ideal α) {x : α} {y : α} (hy : IsUnit y) :

y * x ∈ I ↔ x ∈ I

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def Ideal.span {α : Type u} [Semiring α] (s : Set α) :

Ideal α

The ideal generated by a subset of a ring

Equations

Ideal.span s = Submodule.span α s

Instances For

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@[simp]

theorem Ideal.submodule_span_eq {α : Type u} [Semiring α] {s : Set α} :

Submodule.span α s = Ideal.span s

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@[simp]

theorem Ideal.span_empty {α : Type u} [Semiring α] :

Ideal.span ∅ = ⊥

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@[simp]

theorem Ideal.span_univ {α : Type u} [Semiring α] :

Ideal.span Set.univ = ⊤

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theorem Ideal.span_union {α : Type u} [Semiring α] (s : Set α) (t : Set α) :

Ideal.span (s ∪ t) = Ideal.span s ⊔ Ideal.span t

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theorem Ideal.span_iUnion {α : Type u} [Semiring α] {ι : Sort u_1} (s : ι → Set α) :

Ideal.span (⋃ (i : ι), s i) = ⨆ (i : ι), Ideal.span (s i)

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theorem Ideal.mem_span {α : Type u} [Semiring α] {s : Set α} (x : α) :

x ∈ Ideal.span s ↔ ∀ (p : Ideal α), s ⊆ ↑p → x ∈ p

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theorem Ideal.subset_span {α : Type u} [Semiring α] {s : Set α} :

s ⊆ ↑(Ideal.span s)

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theorem Ideal.span_le {α : Type u} [Semiring α] {s : Set α} {I : Ideal α} :

Ideal.span s ≤ I ↔ s ⊆ ↑I

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theorem Ideal.span_mono {α : Type u} [Semiring α] {s : Set α} {t : Set α} :

s ⊆ t → Ideal.span s ≤ Ideal.span t

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@[simp]

theorem Ideal.span_eq {α : Type u} [Semiring α] (I : Ideal α) :

Ideal.span ↑I = I

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@[simp]

theorem Ideal.span_singleton_one {α : Type u} [Semiring α] :

Ideal.span {1} = ⊤

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theorem Ideal.isCompactElement_top {α : Type u} [Semiring α] :

CompleteLattice.IsCompactElement ⊤

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theorem Ideal.mem_span_insert {α : Type u} [Semiring α] {s : Set α} {x : α} {y : α} :

x ∈ Ideal.span (insert y s) ↔ ∃ (a : α), ∃ z ∈ Ideal.span s, x = a * y + z

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theorem Ideal.mem_span_singleton' {α : Type u} [Semiring α] {x : α} {y : α} :

x ∈ Ideal.span {y} ↔ ∃ (a : α), a * y = x

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theorem Ideal.span_singleton_le_iff_mem {α : Type u} [Semiring α] (I : Ideal α) {x : α} :

Ideal.span {x} ≤ I ↔ x ∈ I

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theorem Ideal.span_singleton_mul_left_unit {α : Type u} [Semiring α] {a : α} (h2 : IsUnit a) (x : α) :

Ideal.span {a * x} = Ideal.span {x}

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theorem Ideal.span_insert {α : Type u} [Semiring α] (x : α) (s : Set α) :

Ideal.span (insert x s) = Ideal.span {x} ⊔ Ideal.span s

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theorem Ideal.span_eq_bot {α : Type u} [Semiring α] {s : Set α} :

Ideal.span s = ⊥ ↔ ∀ x ∈ s, x = 0

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@[simp]

theorem Ideal.span_singleton_eq_bot {α : Type u} [Semiring α] {x : α} :

Ideal.span {x} = ⊥ ↔ x = 0

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theorem Ideal.span_singleton_ne_top {α : Type u_1} [CommSemiring α] {x : α} (hx : ¬IsUnit x) :

Ideal.span {x} ≠ ⊤

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@[simp]

theorem Ideal.span_zero {α : Type u} [Semiring α] :

Ideal.span 0 = ⊥

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@[simp]

theorem Ideal.span_one {α : Type u} [Semiring α] :

Ideal.span 1 = ⊤

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theorem Ideal.span_eq_top_iff_finite {α : Type u} [Semiring α] (s : Set α) :

Ideal.span s = ⊤ ↔ ∃ (s' : Finset α), ↑s' ⊆ s ∧ Ideal.span ↑s' = ⊤

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theorem Ideal.mem_span_singleton_sup {S : Type u_1} [CommSemiring S] {x : S} {y : S} {I : Ideal S} :

x ∈ Ideal.span {y} ⊔ I ↔ ∃ (a : S), ∃ b ∈ I, a * y + b = x

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def Ideal.ofRel {α : Type u} [Semiring α] (r : α → α → Prop) :

Ideal α

The ideal generated by an arbitrary binary relation.

Equations

Ideal.ofRel r = Submodule.span α {x : α | ∃ (a : α) (b : α), r a b ∧ x + b = a}

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class Ideal.IsPrime {α : Type u} [Semiring α] (I : Ideal α) :

Prop

An ideal P of a ring R is prime if P ≠ R and xy ∈ P → x ∈ P ∨ y ∈ P

ne_top' : I ≠ ⊤
The prime ideal is not the entire ring.
mem_or_mem' : ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I
If a product lies in the prime ideal, then at least one element lies in the prime ideal.

Instances

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theorem Ideal.isPrime_iff {α : Type u} [Semiring α] {I : Ideal α} :

Ideal.IsPrime I ↔ I ≠ ⊤ ∧ ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I

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theorem Ideal.IsPrime.ne_top {α : Type u} [Semiring α] {I : Ideal α} (hI : Ideal.IsPrime I) :

I ≠ ⊤

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theorem Ideal.IsPrime.mem_or_mem {α : Type u} [Semiring α] {I : Ideal α} (hI : Ideal.IsPrime I) {x : α} {y : α} :

x * y ∈ I → x ∈ I ∨ y ∈ I

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theorem Ideal.IsPrime.mem_or_mem_of_mul_eq_zero {α : Type u} [Semiring α] {I : Ideal α} (hI : Ideal.IsPrime I) {x : α} {y : α} (h : x * y = 0) :

x ∈ I ∨ y ∈ I

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theorem Ideal.IsPrime.mem_of_pow_mem {α : Type u} [Semiring α] {I : Ideal α} (hI : Ideal.IsPrime I) {r : α} (n : ℕ) (H : r ^ n ∈ I) :

r ∈ I

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theorem Ideal.not_isPrime_iff {α : Type u} [Semiring α] {I : Ideal α} :

¬Ideal.IsPrime I ↔ I = ⊤ ∨ ∃ (x : α) (_ : x ∉ I) (y : α) (_ : y ∉ I), x * y ∈ I

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theorem Ideal.zero_ne_one_of_proper {α : Type u} [Semiring α] {I : Ideal α} (h : I ≠ ⊤) :

0 ≠ 1

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theorem Ideal.bot_prime {α : Type u} [Semiring α] [IsDomain α] :

Ideal.IsPrime ⊥

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class Ideal.IsMaximal {α : Type u} [Semiring α] (I : Ideal α) :

Prop

An ideal is maximal if it is maximal in the collection of proper ideals.

out : IsCoatom I
The maximal ideal is a coatom in the ordering on ideals; that is, it is not the entire ring, and there are no other proper ideals strictly containing it.

Instances

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theorem Ideal.isMaximal_def {α : Type u} [Semiring α] {I : Ideal α} :

Ideal.IsMaximal I ↔ IsCoatom I

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theorem Ideal.IsMaximal.ne_top {α : Type u} [Semiring α] {I : Ideal α} (h : Ideal.IsMaximal I) :

I ≠ ⊤

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theorem Ideal.isMaximal_iff {α : Type u} [Semiring α] {I : Ideal α} :

Ideal.IsMaximal I ↔ 1 ∉ I ∧ ∀ (J : Ideal α) (x : α), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J

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theorem Ideal.IsMaximal.eq_of_le {α : Type u} [Semiring α] {I : Ideal α} {J : Ideal α} (hI : Ideal.IsMaximal I) (hJ : J ≠ ⊤) (IJ : I ≤ J) :

I = J

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instance Ideal.instIsCoatomicIdealToPartialOrderInstOmegaCompletePartialOrderCompleteLatticeToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToModuleInstOrderTopSubmoduleToLEToPreorderInstPartialOrderSetLike {α : Type u} [Semiring α] :

IsCoatomic (Ideal α)

Equations

⋯ = ⋯

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theorem Ideal.IsMaximal.coprime_of_ne {α : Type u} [Semiring α] {M : Ideal α} {M' : Ideal α} (hM : Ideal.IsMaximal M) (hM' : Ideal.IsMaximal M') (hne : M ≠ M') :

M ⊔ M' = ⊤

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theorem Ideal.exists_le_maximal {α : Type u} [Semiring α] (I : Ideal α) (hI : I ≠ ⊤) :

∃ (M : Ideal α), Ideal.IsMaximal M ∧ I ≤ M

Krull's theorem: if I is an ideal that is not the whole ring, then it is included in some maximal ideal.

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theorem Ideal.exists_maximal (α : Type u) [Semiring α] [Nontrivial α] :

∃ (M : Ideal α), Ideal.IsMaximal M

Krull's theorem: a nontrivial ring has a maximal ideal.

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instance Ideal.instNontrivialIdeal {α : Type u} [Semiring α] [Nontrivial α] :

Nontrivial (Ideal α)

Equations

⋯ = ⋯

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theorem Ideal.maximal_of_no_maximal {α : Type u} [Semiring α] {P : Ideal α} (hmax : ∀ (m : Ideal α), P < m → ¬Ideal.IsMaximal m) (J : Ideal α) (hPJ : P < J) :

J = ⊤

If P is not properly contained in any maximal ideal then it is not properly contained in any proper ideal

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theorem Ideal.span_pair_comm {α : Type u} [Semiring α] {x : α} {y : α} :

Ideal.span {x, y} = Ideal.span {y, x}

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theorem Ideal.mem_span_pair {α : Type u} [Semiring α] {x : α} {y : α} {z : α} :

z ∈ Ideal.span {x, y} ↔ ∃ (a : α) (b : α), a * x + b * y = z

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@[simp]

theorem Ideal.span_pair_add_mul_left {R : Type u} [CommRing R] {x : R} {y : R} (z : R) :

Ideal.span {x + y * z, y} = Ideal.span {x, y}

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@[simp]

theorem Ideal.span_pair_add_mul_right {R : Type u} [CommRing R] {x : R} {y : R} (z : R) :

Ideal.span {x, y + x * z} = Ideal.span {x, y}

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theorem Ideal.IsMaximal.exists_inv {α : Type u} [Semiring α] {I : Ideal α} (hI : Ideal.IsMaximal I) {x : α} (hx : x ∉ I) :

∃ (y : α), ∃ i ∈ I, y * x + i = 1

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theorem Ideal.mem_sup_left {R : Type u} [Semiring R] {S : Ideal R} {T : Ideal R} {x : R} :

x ∈ S → x ∈ S ⊔ T

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theorem Ideal.mem_sup_right {R : Type u} [Semiring R] {S : Ideal R} {T : Ideal R} {x : R} :

x ∈ T → x ∈ S ⊔ T

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theorem Ideal.mem_iSup_of_mem {R : Type u} [Semiring R] {ι : Sort u_1} {S : ι → Ideal R} (i : ι) {x : R} :

x ∈ S i → x ∈ iSup S

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theorem Ideal.mem_sSup_of_mem {R : Type u} [Semiring R] {S : Set (Ideal R)} {s : Ideal R} (hs : s ∈ S) {x : R} :

x ∈ s → x ∈ sSup S

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theorem Ideal.mem_sInf {R : Type u} [Semiring R] {s : Set (Ideal R)} {x : R} :

x ∈ sInf s ↔ ∀ ⦃I : Ideal R⦄, I ∈ s → x ∈ I

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@[simp]

theorem Ideal.mem_inf {R : Type u} [Semiring R] {I : Ideal R} {J : Ideal R} {x : R} :

x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J

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@[simp]

theorem Ideal.mem_iInf {R : Type u} [Semiring R] {ι : Sort u_1} {I : ι → Ideal R} {x : R} :

x ∈ iInf I ↔ ∀ (i : ι), x ∈ I i

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@[simp]

theorem Ideal.mem_bot {R : Type u} [Semiring R] {x : R} :

x ∈ ⊥ ↔ x = 0

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def Ideal.pi {α : Type u} [Semiring α] (I : Ideal α) (ι : Type v) :

Ideal (ι → α)

I^n as an ideal of R^n.

Equations

Ideal.pi I ι = { toAddSubmonoid := { toAddSubsemigroup := { carrier := {x : ι → α | ∀ (i : ι), x i ∈ I}, add_mem' := ⋯ }, zero_mem' := ⋯ }, smul_mem' := ⋯ }

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theorem Ideal.mem_pi {α : Type u} [Semiring α] (I : Ideal α) (ι : Type v) (x : ι → α) :

x ∈ Ideal.pi I ι ↔ ∀ (i : ι), x i ∈ I

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theorem Ideal.sInf_isPrime_of_isChain {α : Type u} [Semiring α] {s : Set (Ideal α)} (hs : Set.Nonempty s) (hs' : IsChain (fun (x x_1 : Ideal α) => x ≤ x_1) s) (H : ∀ p ∈ s, Ideal.IsPrime p) :

Ideal.IsPrime (sInf s)

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@[simp]

theorem Ideal.mul_unit_mem_iff_mem {α : Type u} [CommSemiring α] (I : Ideal α) {x : α} {y : α} (hy : IsUnit y) :

x * y ∈ I ↔ x ∈ I

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theorem Ideal.mem_span_singleton {α : Type u} [CommSemiring α] {x : α} {y : α} :

x ∈ Ideal.span {y} ↔ y ∣ x

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theorem Ideal.mem_span_singleton_self {α : Type u} [CommSemiring α] (x : α) :

x ∈ Ideal.span {x}

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theorem Ideal.span_singleton_le_span_singleton {α : Type u} [CommSemiring α] {x : α} {y : α} :

Ideal.span {x} ≤ Ideal.span {y} ↔ y ∣ x

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theorem Ideal.span_singleton_eq_span_singleton {α : Type u} [CommRing α] [IsDomain α] {x : α} {y : α} :

Ideal.span {x} = Ideal.span {y} ↔ Associated x y

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theorem Ideal.span_singleton_mul_right_unit {α : Type u} [CommSemiring α] {a : α} (h2 : IsUnit a) (x : α) :

Ideal.span {x * a} = Ideal.span {x}

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@[simp]

theorem Ideal.span_singleton_eq_top {α : Type u} [CommSemiring α] {x : α} :

Ideal.span {x} = ⊤ ↔ IsUnit x

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theorem Ideal.span_singleton_prime {α : Type u} [CommSemiring α] {p : α} (hp : p ≠ 0) :

Ideal.IsPrime (Ideal.span {p}) ↔ Prime p

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theorem Ideal.IsMaximal.isPrime {α : Type u} [CommSemiring α] {I : Ideal α} (H : Ideal.IsMaximal I) :

Ideal.IsPrime I

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instance Ideal.IsMaximal.isPrime' {α : Type u} [CommSemiring α] (I : Ideal α) [_H : Ideal.IsMaximal I] :

Ideal.IsPrime I

Equations

⋯ = ⋯

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theorem Ideal.span_singleton_lt_span_singleton {α : Type u} [CommSemiring α] [IsDomain α] {x : α} {y : α} :

Ideal.span {x} < Ideal.span {y} ↔ DvdNotUnit y x

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theorem Ideal.factors_decreasing {α : Type u} [CommSemiring α] [IsDomain α] (b₁ : α) (b₂ : α) (h₁ : b₁ ≠ 0) (h₂ : ¬IsUnit b₂) :

Ideal.span {b₁ * b₂} < Ideal.span {b₁}

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theorem Ideal.mul_mem_right {α : Type u} {a : α} (b : α) [CommSemiring α] (I : Ideal α) (h : a ∈ I) :

a * b ∈ I

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theorem Ideal.pow_mem_of_mem {α : Type u} {a : α} [CommSemiring α] (I : Ideal α) (ha : a ∈ I) (n : ℕ) (hn : 0 < n) :

a ^ n ∈ I

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theorem Ideal.IsPrime.mul_mem_iff_mem_or_mem {α : Type u} [CommSemiring α] {I : Ideal α} (hI : Ideal.IsPrime I) {x : α} {y : α} :

x * y ∈ I ↔ x ∈ I ∨ y ∈ I

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theorem Ideal.IsPrime.pow_mem_iff_mem {α : Type u} [CommSemiring α] {I : Ideal α} (hI : Ideal.IsPrime I) {r : α} (n : ℕ) (hn : 0 < n) :

r ^ n ∈ I ↔ r ∈ I

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theorem Ideal.pow_multiset_sum_mem_span_pow {α : Type u} [CommSemiring α] [DecidableEq α] (s : Multiset α) (n : ℕ) :

Multiset.sum s ^ (Multiset.card s * n + 1) ∈ Ideal.span ↑(Multiset.toFinset (Multiset.map (fun (x : α) => x ^ (n + 1)) s))

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theorem Ideal.sum_pow_mem_span_pow {α : Type u} [CommSemiring α] {ι : Type u_1} (s : Finset ι) (f : ι → α) (n : ℕ) :

(Finset.sum s fun (i : ι) => f i) ^ (s.card * n + 1) ∈ Ideal.span ((fun (i : ι) => f i ^ (n + 1)) '' ↑s)

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theorem Ideal.span_pow_eq_top {α : Type u} [CommSemiring α] (s : Set α) (hs : Ideal.span s = ⊤) (n : ℕ) :

Ideal.span ((fun (x : α) => x ^ n) '' s) = ⊤

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theorem Ideal.isPrime_of_maximally_disjoint {α : Type u} [CommSemiring α] (I : Ideal α) (S : Submonoid α) (disjoint : Disjoint ↑I ↑S) (maximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S) :

Ideal.IsPrime I

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