mathlib documentation

ring_theory.ideal.basic

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.

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@[protected]
theorem ideal.zero_mem {α : Type u} [semiring α] (I : ideal α) :
0 ∈ I
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@[protected]
theorem ideal.add_mem {α : Type u} [semiring α] (I : ideal α) {a 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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@[ext]
theorem ideal.ext {α : Type u} [semiring α] {I 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 : ι), c ∈ t → f c ∈ I) → t.sum (λ (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 y : α) (hx : x ∈ I) (h : y * x = 1) :
I = ⊤
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theorem ideal.eq_top_of_is_unit_mem {α : Type u} [semiring α] (I : ideal α) {x : α} (hx : x ∈ I) (h : is_unit 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 y : α} (hy : is_unit 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

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Instances for ideal.span
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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 t : set α) :
ideal.span (s ∪ t) = ideal.span s ⊔ ideal.span t
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theorem ideal.span_Union {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 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.mem_span_insert {α : Type u} [semiring α] {s : set α} {x y : α} :
x ∈ ideal.span (has_insert.insert y s) ↔ ∃ (a z : α) (H : z ∈ ideal.span s), x = a * y + z
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theorem ideal.mem_span_singleton' {α : Type u} [semiring α] {x y : α} :
x ∈ ideal.span {y} ↔ ∃ (a : α), a * y = x
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theorem ideal.span_insert {α : Type u} [semiring α] (x : α) (s : set α) :
ideal.span (has_insert.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 : α), 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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@[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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def ideal.of_rel {α : Type u} [semiring α] (r : α → α → Prop) :
ideal α

The ideal generated by an arbitrary binary relation.

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@[class]
structure ideal.is_prime {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

Instances of this typeclass
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theorem ideal.is_prime_iff {α : Type u} [semiring α] {I : ideal α} :
I.is_prime ↔ I ≠ ⊤ ∧ ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I
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theorem ideal.is_prime.ne_top {α : Type u} [semiring α] {I : ideal α} (hI : I.is_prime) :
I ≠ ⊤
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theorem ideal.is_prime.mem_or_mem {α : Type u} [semiring α] {I : ideal α} (hI : I.is_prime) {x y : α} :
x * y ∈ I → x ∈ I ∨ y ∈ I
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theorem ideal.is_prime.mem_or_mem_of_mul_eq_zero {α : Type u} [semiring α] {I : ideal α} (hI : I.is_prime) {x y : α} (h : x * y = 0) :
x ∈ I ∨ y ∈ I
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theorem ideal.is_prime.mem_of_pow_mem {α : Type u} [semiring α] {I : ideal α} (hI : I.is_prime) {r : α} (n : ℕ) (H : r ^ n ∈ I) :
r ∈ I
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theorem ideal.not_is_prime_iff {α : Type u} [semiring α] {I : ideal α} :
¬I.is_prime ↔ I = ⊤ ∨ ∃ (x : α) (H : x ∉ I) (y : α) (H : 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 {R : Type u_1} [ring R] [is_domain R] :
⊥.is_prime
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@[class]
structure ideal.is_maximal {α : Type u} [semiring α] (I : ideal α) :
Prop

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

Instances of this typeclass
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theorem ideal.is_maximal_def {α : Type u} [semiring α] {I : ideal α} :
I.is_maximal ↔ is_coatom I
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theorem ideal.is_maximal.ne_top {α : Type u} [semiring α] {I : ideal α} (h : I.is_maximal) :
I ≠ ⊤
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theorem ideal.is_maximal_iff {α : Type u} [semiring α] {I : ideal α} :
I.is_maximal ↔ 1 ∉ I ∧ ∀ (J : ideal α) (x : α), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J
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theorem ideal.is_maximal.eq_of_le {α : Type u} [semiring α] {I J : ideal α} (hI : I.is_maximal) (hJ : J ≠ ⊤) (IJ : I ≤ J) :
I = J
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@[protected, instance]
def ideal.is_coatomic {α : Type u} [semiring α] :
is_coatomic (ideal α)
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theorem ideal.exists_le_maximal {α : Type u} [semiring α] (I : ideal α) (hI : I ≠ ⊤) :
∃ (M : ideal α), M.is_maximal ∧ 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), M.is_maximal

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

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@[protected, instance]
def ideal.nontrivial {α : Type u} [semiring α] [nontrivial α] :
nontrivial (ideal α)
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theorem ideal.maximal_of_no_maximal {R : Type u} [semiring R] {P : ideal R} (hmax : ∀ (m : ideal R), P < m → ¬m.is_maximal) (J : ideal R) (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.mem_span_pair {α : Type u} [semiring α] {x y z : α} :
z ∈ ideal.span {x, y} ↔ ∃ (a b : α), a * x + b * y = z
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theorem ideal.is_maximal.exists_inv {α : Type u} [semiring α] {I : ideal α} (hI : I.is_maximal) {x : α} (hx : x ∉ I) :
∃ (y i : α) (H : i ∈ I), y * x + i = 1
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theorem ideal.mem_sup_left {R : Type u} [semiring R] {S 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 T : ideal R} {x : R} :
x ∈ T → x ∈ S ⊔ T
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theorem ideal.mem_supr_of_mem {R : Type u} [semiring R] {ι : Sort u_1} {S : ι → ideal R} (i : ι) {x : R} :
x ∈ S i → x ∈ supr S
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theorem ideal.mem_Sup_of_mem {R : Type u} [semiring R] {S : set (ideal R)} {s : ideal R} (hs : s ∈ S) {x : R} :
x ∈ s → x ∈ has_Sup.Sup S
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theorem ideal.mem_Inf {R : Type u} [semiring R] {s : set (ideal R)} {x : R} :
x ∈ has_Inf.Inf s ↔ ∀ ⦃I : ideal R⦄, I ∈ s → x ∈ I
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@[simp]
theorem ideal.mem_inf {R : Type u} [semiring R] {I J : ideal R} {x : R} :
x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J
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@[simp]
theorem ideal.mem_infi {R : Type u} [semiring R] {ι : Sort u_1} {I : ι → ideal R} {x : R} :
x ∈ infi 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
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theorem ideal.mem_pi {α : Type u} [semiring α] (I : ideal α) (ι : Type v) (x : ι → α) :
x ∈ I.pi ι ↔ ∀ (i : ι), x i ∈ I
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@[simp]
theorem ideal.mul_unit_mem_iff_mem {α : Type u} [comm_semiring α] (I : ideal α) {x y : α} (hy : is_unit y) :
x * y ∈ I ↔ x ∈ I
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theorem ideal.mem_span_singleton {α : Type u} [comm_semiring α] {x y : α} :
x ∈ ideal.span {y} ↔ y ∣ x
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theorem ideal.span_singleton_le_span_singleton {α : Type u} [comm_semiring α] {x y : α} :
ideal.span {x} ≤ ideal.span {y} ↔ y ∣ x
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theorem ideal.span_singleton_eq_span_singleton {α : Type u} [comm_ring α] [is_domain α] {x 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} [comm_semiring α] {a : α} (h2 : is_unit a) (x : α) :
ideal.span {x * a} = ideal.span {x}
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theorem ideal.span_singleton_mul_left_unit {α : Type u} [comm_semiring α] {a : α} (h2 : is_unit a) (x : α) :
ideal.span {a * x} = ideal.span {x}
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theorem ideal.span_singleton_eq_top {α : Type u} [comm_semiring α] {x : α} :
ideal.span {x} = ⊤ ↔ is_unit x
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theorem ideal.span_singleton_prime {α : Type u} [comm_semiring α] {p : α} (hp : p ≠ 0) :
(ideal.span {p}).is_prime ↔ prime p
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theorem ideal.is_maximal.is_prime {α : Type u} [comm_semiring α] {I : ideal α} (H : I.is_maximal) :
I.is_prime
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@[protected, instance]
def ideal.is_maximal.is_prime' {α : Type u} [comm_semiring α] (I : ideal α) [H : I.is_maximal] :
I.is_prime
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theorem ideal.span_singleton_lt_span_singleton {β : Type v} [comm_ring β] [is_domain β] {x y : β} :
ideal.span {x} < ideal.span {y} ↔ dvd_not_unit y x
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theorem ideal.factors_decreasing {β : Type v} [comm_ring β] [is_domain β] (b₁ b₂ : β) (h₁ : b₁ ≠ 0) (h₂ : ¬is_unit b₂) :
ideal.span {b₁ * b₂} < ideal.span {b₁}
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theorem ideal.mul_mem_right {α : Type u} {a : α} (b : α) [comm_semiring 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} [comm_semiring α] (I : ideal α) (ha : a ∈ I) (n : ℕ) (hn : 0 < n) :
a ^ n ∈ I
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theorem ideal.is_prime.mul_mem_iff_mem_or_mem {α : Type u} [comm_semiring α] {I : ideal α} (hI : I.is_prime) {x y : α} :
x * y ∈ I ↔ x ∈ I ∨ y ∈ I
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theorem ideal.is_prime.pow_mem_iff_mem {α : Type u} [comm_semiring α] {I : ideal α} (hI : I.is_prime) {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} [comm_semiring α] (s : multiset α) (n : ℕ) :
s.sum ^ (⇑multiset.card s * n + 1) ∈ ideal.span ↑((multiset.map (λ (x : α), x ^ (n + 1)) s).to_finset)
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theorem ideal.sum_pow_mem_span_pow {α : Type u} [comm_semiring α] {ι : Type u_1} (s : finset ι) (f : ι → α) (n : ℕ) :
s.sum (λ (i : ι), f i) ^ (s.card * n + 1) ∈ ideal.span ((λ (i : ι), f i ^ (n + 1)) '' ↑s)
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theorem ideal.span_pow_eq_top {α : Type u} [comm_semiring α] (s : set α) (hs : ideal.span s = ⊤) (n : ℕ) :
ideal.span ((λ (x : α), x ^ n) '' s) = ⊤
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@[protected]
theorem ideal.neg_mem_iff {α : Type u} [ring α] (I : ideal α) {a : α} :
-a ∈ I ↔ a ∈ I
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@[protected]
theorem ideal.add_mem_iff_left {α : Type u} [ring α] (I : ideal α) {a b : α} :
b ∈ I → (a + b ∈ I ↔ a ∈ I)
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@[protected]
theorem ideal.add_mem_iff_right {α : Type u} [ring α] (I : ideal α) {a b : α} :
a ∈ I → (a + b ∈ I ↔ b ∈ I)
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@[protected]
theorem ideal.sub_mem {α : Type u} [ring α] (I : ideal α) {a b : α} :
a ∈ I → b ∈ I → a - b ∈ I
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theorem ideal.mem_span_insert' {α : Type u} [ring α] {s : set α} {x y : α} :
x ∈ ideal.span (has_insert.insert y s) ↔ ∃ (a : α), x + a * y ∈ ideal.span s
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theorem ideal.eq_bot_or_top {K : Type u} [division_ring K] (I : ideal K) :
I = ⊥ ∨ I = ⊤

All ideals in a division ring are trivial.

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theorem ideal.eq_bot_of_prime {K : Type u} [division_ring K] (I : ideal K) [h : I.is_prime] :
I = ⊥
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theorem ideal.bot_is_maximal {K : Type u} [division_ring K] :
⊥.is_maximal
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theorem ideal.mul_sub_mul_mem {R : Type u_1} [comm_ring R] (I : ideal R) {a b c d : R} (h1 : a - b ∈ I) (h2 : c - d ∈ I) :
a * c - b * d ∈ I
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theorem ring.not_is_field_of_subsingleton {R : Type u_1} [ring R] [subsingleton R] :
¬is_field R
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theorem ring.exists_not_is_unit_of_not_is_field {R : Type u_1} [comm_ring R] [nontrivial R] (hf : ¬is_field R) :
∃ (x : R) (H : x ≠ 0), ¬is_unit x
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theorem ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top {R : Type u_1} [comm_ring R] [nontrivial R] :
¬is_field R ↔ ∃ (I : ideal R), ⊥ < I ∧ I < ⊤
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theorem ring.not_is_field_iff_exists_prime {R : Type u_1} [comm_ring R] [nontrivial R] :
¬is_field R ↔ ∃ (p : ideal R), p ≠ ⊥ ∧ p.is_prime
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theorem ring.ne_bot_of_is_maximal_of_not_is_field {R : Type u_1} [comm_ring R] [nontrivial R] {M : ideal R} (max : M.is_maximal) (not_field : ¬is_field R) :
M ≠ ⊥

When a ring is not a field, the maximal ideals are nontrivial.

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theorem ideal.bot_lt_of_maximal {R : Type u} [comm_ring R] [nontrivial R] (M : ideal R) [hm : M.is_maximal] (non_field : ¬is_field R) :
⊥ < M
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def nonunits (α : Type u) [monoid α] :
set α

The set of non-invertible elements of a monoid.

Equations
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@[simp]
theorem mem_nonunits_iff {α : Type u} {a : α} [monoid α] :
a ∈ nonunits α ↔ ¬is_unit a
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theorem mul_mem_nonunits_right {α : Type u} {a b : α} [comm_monoid α] :
b ∈ nonunits α → a * b ∈ nonunits α
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theorem mul_mem_nonunits_left {α : Type u} {a b : α} [comm_monoid α] :
a ∈ nonunits α → a * b ∈ nonunits α
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theorem zero_mem_nonunits {α : Type u} [semiring α] :
0 ∈ nonunits α ↔ 0 ≠ 1
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@[simp]
theorem one_not_mem_nonunits {α : Type u} [monoid α] :
1 ∉ nonunits α
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theorem coe_subset_nonunits {α : Type u} [semiring α] {I : ideal α} (h : I ≠ ⊤) :
↑I ⊆ nonunits α
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theorem exists_max_ideal_of_mem_nonunits {α : Type u} {a : α} [comm_semiring α] (h : a ∈ nonunits α) :
∃ (I : ideal α), I.is_maximal ∧ a ∈ I