statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
neg_mem_iff : -a ∈ I ↔ a ∈ I | neg_mem_iff | lemma | ideal.neg_mem_iff | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_iff_left : b ∈ I → (a + b ∈ I ↔ a ∈ I) | I.add_mem_iff_left | lemma | ideal.add_mem_iff_left | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_iff_right : a ∈ I → (a + b ∈ I ↔ b ∈ I) | I.add_mem_iff_right | lemma | ideal.add_mem_iff_right | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mem : a ∈ I → b ∈ I → a - b ∈ I | sub_mem | lemma | ideal.sub_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_insert' {s : set α} {x y} :
x ∈ span (insert y s) ↔ ∃a, x + a * y ∈ span s | submodule.mem_span_insert' | lemma | ideal.mem_span_insert' | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.mem_span_insert'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_neg (x : α) : (span {-x} : ideal α) = span {x} | by { ext, simp only [mem_span_singleton'],
exact ⟨λ ⟨y, h⟩, ⟨-y, h ▸ neg_mul_comm y x⟩, λ ⟨y, h⟩, ⟨-y, h ▸ neg_mul_neg y x⟩⟩ } | lemma | ideal.span_singleton_neg | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"neg_mul_comm",
"neg_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_or_top : I = ⊥ ∨ I = ⊤ | begin
rw or_iff_not_imp_right,
change _ ≠ _ → _,
rw ideal.ne_top_iff_one,
intro h1,
rw eq_bot_iff,
intros r hr,
by_cases H : r = 0, {simpa},
simpa [H, h1] using I.mul_mem_left r⁻¹ hr,
end | lemma | ideal.eq_bot_or_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"eq_bot_iff",
"ideal.ne_top_iff_one",
"or_iff_not_imp_right"
] | All ideals in a division (semi)ring are trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_bot_of_prime [h : I.is_prime] : I = ⊥ | or_iff_not_imp_right.mp I.eq_bot_or_top h.1 | lemma | ideal.eq_bot_of_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_is_maximal : is_maximal (⊥ : ideal K) | ⟨⟨λ h, absurd ((eq_top_iff_one (⊤ : ideal K)).mp rfl) (by rw ← h; simp),
λ I hI, or_iff_not_imp_left.mp (eq_bot_or_top I) (ne_of_gt hI)⟩⟩ | lemma | ideal.bot_is_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_mul_mem {R : Type*} [comm_ring R] (I : ideal R) {a b c d : R}
(h1 : a - b ∈ I) (h2 : c - d ∈ I) : a * c - b * d ∈ I | begin
rw (show a * c - b * d = (a - b) * c + b * (c - d), by {rw [sub_mul, mul_sub], abel}),
exact I.add_mem (I.mul_mem_right _ h1) (I.mul_mem_left _ h2),
end | theorem | ideal.mul_sub_mul_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_not_is_unit_of_not_is_field [nontrivial R] (hf : ¬ is_field R) :
∃ x ≠ (0 : R), ¬ is_unit x | begin
have : ¬ _ := λ h, hf ⟨exists_pair_ne R, mul_comm, h⟩,
simp_rw is_unit_iff_exists_inv,
push_neg at ⊢ this,
obtain ⟨x, hx, not_unit⟩ := this,
exact ⟨x, hx, not_unit⟩
end | lemma | ring.exists_not_is_unit_of_not_is_field | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_field",
"is_unit",
"is_unit_iff_exists_inv",
"mul_comm",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_field_iff_exists_ideal_bot_lt_and_lt_top [nontrivial R] :
¬ is_field R ↔ ∃ I : ideal R, ⊥ < I ∧ I < ⊤ | begin
split,
{ intro h,
obtain ⟨x, nz, nu⟩ := exists_not_is_unit_of_not_is_field h,
use ideal.span {x},
rw [bot_lt_iff_ne_bot, lt_top_iff_ne_top],
exact ⟨mt ideal.span_singleton_eq_bot.mp nz, mt ideal.span_singleton_eq_top.mp nu⟩ },
{ rintros ⟨I, bot_lt, lt_top⟩ hf,
obtain ⟨x, mem, ne_zero⟩ :=... | lemma | ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"bot_lt_iff_ne_bot",
"ideal",
"ideal.eq_top_iff_one",
"ideal.span",
"is_field",
"lt_top_iff_ne_top",
"ne_zero",
"nontrivial",
"set_like.exists_of_lt",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_field_iff_exists_prime [nontrivial R] :
¬ is_field R ↔ ∃ p : ideal R, p ≠ ⊥ ∧ p.is_prime | not_is_field_iff_exists_ideal_bot_lt_and_lt_top.trans
⟨λ ⟨I, bot_lt, lt_top⟩, let ⟨p, hp, le_p⟩ := I.exists_le_maximal (lt_top_iff_ne_top.mp lt_top) in
⟨p, bot_lt_iff_ne_bot.mp (lt_of_lt_of_le bot_lt le_p), hp.is_prime⟩,
λ ⟨p, ne_bot, prime⟩, ⟨p, bot_lt_iff_ne_bot.mpr ne_bot, lt_top_iff_ne_top.mpr prime.1⟩⟩ | lemma | ring.not_is_field_iff_exists_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"is_field",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field_iff_is_simple_order_ideal :
is_field R ↔ is_simple_order (ideal R) | begin
casesI subsingleton_or_nontrivial R,
{ exact ⟨λ h, (not_is_field_of_subsingleton _ h).elim,
λ h, by exactI (false_of_nontrivial_of_subsingleton $ ideal R).elim⟩ },
rw [← not_iff_not, ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top, ← not_iff_not],
push_neg,
simp_rw [lt_top_iff_ne_top, bot_lt_... | lemma | ring.is_field_iff_is_simple_order_ideal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"bot_lt_iff_ne_bot",
"false_of_nontrivial_of_subsingleton",
"ideal",
"is_field",
"is_simple_order",
"lt_top_iff_ne_top",
"not_iff_not",
"not_is_field_of_subsingleton",
"not_ne_iff",
"or_iff_not_imp_left",
"ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top",
"subsingleton_or_nontrivial"
] | Also see `ideal.is_simple_order` for the forward direction as an instance when `R` is a
division (semi)ring.
This result actually holds for all division semirings, but we lack the predicate to state it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_bot_of_is_maximal_of_not_is_field [nontrivial R] {M : ideal R} (max : M.is_maximal)
(not_field : ¬ is_field R) : M ≠ ⊥ | begin
rintros h,
rw h at max,
rcases max with ⟨⟨h1, h2⟩⟩,
obtain ⟨I, hIbot, hItop⟩ := not_is_field_iff_exists_ideal_bot_lt_and_lt_top.mp not_field,
exact ne_of_lt hItop (h2 I hIbot),
end | lemma | ring.ne_bot_of_is_maximal_of_not_is_field | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"is_field",
"nontrivial"
] | When a ring is not a field, the maximal ideals are nontrivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_lt_of_maximal (M : ideal R) [hm : M.is_maximal] (non_field : ¬ is_field R) : ⊥ < M | begin
rcases (ring.not_is_field_iff_exists_ideal_bot_lt_and_lt_top.1 non_field)
with ⟨I, Ibot, Itop⟩,
split, { simp },
intro mle,
apply @irrefl _ (<) _ (⊤ : ideal R),
have : M = ⊥ := eq_bot_iff.mpr mle,
rw this at *,
rwa hm.1.2 I Ibot at Itop,
end | lemma | ideal.bot_lt_of_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"is_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits (α : Type u) [monoid α] : set α | { a | ¬is_unit a } | def | nonunits | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit",
"monoid"
] | The set of non-invertible elements of a monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nonunits_iff [monoid α] : a ∈ nonunits α ↔ ¬ is_unit a | iff.rfl | theorem | mem_nonunits_iff | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit",
"monoid",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_nonunits_right [comm_monoid α] :
b ∈ nonunits α → a * b ∈ nonunits α | mt is_unit_of_mul_is_unit_right | theorem | mul_mem_nonunits_right | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_monoid",
"is_unit_of_mul_is_unit_right",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_nonunits_left [comm_monoid α] :
a ∈ nonunits α → a * b ∈ nonunits α | mt is_unit_of_mul_is_unit_left | theorem | mul_mem_nonunits_left | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_monoid",
"is_unit_of_mul_is_unit_left",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_nonunits [semiring α] : 0 ∈ nonunits α ↔ (0:α) ≠ 1 | not_congr is_unit_zero_iff | theorem | zero_mem_nonunits | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit_zero_iff",
"nonunits",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_not_mem_nonunits [monoid α] : (1:α) ∉ nonunits α | not_not_intro is_unit_one | theorem | one_not_mem_nonunits | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit_one",
"monoid",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subset_nonunits [semiring α] {I : ideal α} (h : I ≠ ⊤) :
(I : set α) ⊆ nonunits α | λ x hx hu, h $ I.eq_top_of_is_unit_mem hx hu | theorem | coe_subset_nonunits | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"nonunits",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_max_ideal_of_mem_nonunits [comm_semiring α] (h : a ∈ nonunits α) :
∃ I : ideal α, I.is_maximal ∧ a ∈ I | begin
have : ideal.span ({a} : set α) ≠ ⊤,
{ intro H, rw ideal.span_singleton_eq_top at H, contradiction },
rcases ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩,
use [I, Imax], apply H, apply ideal.subset_span, exact set.mem_singleton a
end | lemma | exists_max_ideal_of_mem_nonunits | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_semiring",
"ideal",
"ideal.exists_le_maximal",
"ideal.span",
"ideal.span_singleton_eq_top",
"ideal.subset_span",
"nonunits",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent : Type* | I ⧸ (I • ⊤ : submodule R I) | def | ideal.cotangent | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"submodule"
] | `I ⧸ I ^ 2` as a quotient of `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotangent.module_of_tower : module S I.cotangent | submodule.quotient.module' _ | instance | ideal.cotangent.module_of_tower | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"module",
"submodule.quotient.module'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent : I →ₗ[R] I.cotangent | submodule.mkq _ | def | ideal.to_cotangent | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"submodule.mkq"
] | The quotient map from `I` to `I ⧸ I ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_to_cotangent_ker : I.to_cotangent.ker.map I.subtype = I ^ 2 | by simp [ideal.to_cotangent, submodule.map_smul'', pow_two] | lemma | ideal.map_to_cotangent_ker | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"ideal.to_cotangent",
"pow_two",
"submodule.map_smul''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_cotangent_ker {x : I} : x ∈ I.to_cotangent.ker ↔ (x : R) ∈ I ^ 2 | begin
rw ← I.map_to_cotangent_ker,
simp,
end | lemma | ideal.mem_to_cotangent_ker | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent_eq {x y : I} : I.to_cotangent x = I.to_cotangent y ↔ (x - y : R) ∈ I ^ 2 | begin
rw [← sub_eq_zero, ← map_sub],
exact I.mem_to_cotangent_ker
end | lemma | ideal.to_cotangent_eq | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent_eq_zero (x : I) : I.to_cotangent x = 0 ↔ (x : R) ∈ I ^ 2 | I.mem_to_cotangent_ker | lemma | ideal.to_cotangent_eq_zero | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent_surjective : function.surjective I.to_cotangent | submodule.mkq_surjective _ | lemma | ideal.to_cotangent_surjective | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"submodule.mkq_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent_range : I.to_cotangent.range = ⊤ | submodule.range_mkq _ | lemma | ideal.to_cotangent_range | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"submodule.range_mkq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent_subsingleton_iff :
subsingleton I.cotangent ↔ is_idempotent_elem I | begin
split,
{ introI H,
refine (pow_two I).symm.trans (le_antisymm (ideal.pow_le_self two_ne_zero) _),
exact λ x hx, (I.to_cotangent_eq_zero ⟨x, hx⟩).mp (subsingleton.elim _ _) },
{ exact λ e, ⟨λ x y, quotient.induction_on₂' x y $ λ x y,
I.to_cotangent_eq.mpr $ ((pow_two I).trans e).symm ▸ I.sub_me... | lemma | ideal.cotangent_subsingleton_iff | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"ideal.pow_le_self",
"is_idempotent_elem",
"pow_two",
"quotient.induction_on₂'",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent_to_quotient_square : I.cotangent →ₗ[R] R ⧸ I ^ 2 | submodule.mapq (I • ⊤) (I ^ 2) I.subtype
(by { rw [← submodule.map_le_iff_le_comap, submodule.map_smul'', submodule.map_top,
submodule.range_subtype, smul_eq_mul, pow_two], exact rfl.le }) | def | ideal.cotangent_to_quotient_square | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"pow_two",
"smul_eq_mul",
"submodule.map_le_iff_le_comap",
"submodule.map_smul''",
"submodule.map_top",
"submodule.mapq",
"submodule.range_subtype"
] | The inclusion map `I ⧸ I ^ 2` to `R ⧸ I ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_quotient_square_comp_to_cotangent : I.cotangent_to_quotient_square.comp I.to_cotangent =
(I ^ 2).mkq.comp (submodule.subtype I) | linear_map.ext $ λ _, rfl | lemma | ideal.to_quotient_square_comp_to_cotangent | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"linear_map.ext",
"submodule.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cotangent_to_quotient_square (x : I) : I.cotangent_to_quotient_square (I.to_cotangent x) =
(I ^ 2).mkq x | rfl | lemma | ideal.to_cotangent_to_quotient_square | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent_ideal (I : ideal R) : ideal (R ⧸ I ^ 2) | begin
haveI : @ring_hom_surjective R (R ⧸ I ^ 2) _ _ _ := ⟨ideal.quotient.mk_surjective⟩,
let rq := (I ^ 2)^.quotient.mk,
exact submodule.map rq.to_semilinear_map I,
end | def | ideal.cotangent_ideal | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"ideal",
"ring_hom_surjective",
"submodule.map"
] | `I ⧸ I ^ 2` as an ideal of `R ⧸ I ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotangent_ideal_square (I : ideal R) : I.cotangent_ideal ^ 2 = ⊥ | begin
rw [eq_bot_iff, pow_two I.cotangent_ideal, ← smul_eq_mul],
intros x hx,
apply submodule.smul_induction_on hx,
{ rintros _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩, apply (submodule.quotient.eq _).mpr _,
rw [sub_zero, pow_two], exact ideal.mul_mem_mul hx hy },
{ intros x y hx hy, exact add_mem hx hy }
end | lemma | ideal.cotangent_ideal_square | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"eq_bot_iff",
"ideal",
"ideal.mul_mem_mul",
"pow_two",
"smul_eq_mul",
"submodule.quotient.eq",
"submodule.smul_induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quotient_square_range :
I.cotangent_to_quotient_square.range = I.cotangent_ideal.restrict_scalars R | begin
transitivity (I.cotangent_to_quotient_square.comp I.to_cotangent).range,
{ rw [linear_map.range_comp, I.to_cotangent_range, submodule.map_top] },
{ rw [to_quotient_square_comp_to_cotangent, linear_map.range_comp, I.range_subtype], ext, refl }
end | lemma | ideal.to_quotient_square_range | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"linear_map.range_comp",
"submodule.map_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent_equiv_ideal : I.cotangent ≃ₗ[R] I.cotangent_ideal | begin
refine
{ ..(I.cotangent_to_quotient_square.cod_restrict (I.cotangent_ideal.restrict_scalars R)
(λ x, by { rw ← to_quotient_square_range, exact linear_map.mem_range_self _ _ })),
..(equiv.of_bijective _ ⟨_, _⟩) },
{ rintros x y e,
replace e := congr_arg subtype.val e,
obtain ⟨x, rfl⟩ := I.to_... | def | ideal.cotangent_equiv_ideal | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"equiv.of_bijective",
"linear_map.mem_range_self",
"submodule.mkq_apply",
"submodule.quotient.eq",
"subtype.ext"
] | The equivalence of the two definitions of `I / I ^ 2`, either as the quotient of `I` or the
ideal of `R / I ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotangent_equiv_ideal_apply (x : I.cotangent) :
↑(I.cotangent_equiv_ideal x) = I.cotangent_to_quotient_square x | rfl | lemma | ideal.cotangent_equiv_ideal_apply | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotangent_equiv_ideal_symm_apply (x : R) (hx : x ∈ I) :
I.cotangent_equiv_ideal.symm ⟨(I ^ 2).mkq x, submodule.mem_map_of_mem hx⟩ =
I.to_cotangent ⟨x, hx⟩ | begin
apply I.cotangent_equiv_ideal.injective,
rw I.cotangent_equiv_ideal.apply_symm_apply,
ext,
refl
end | lemma | ideal.cotangent_equiv_ideal_symm_apply | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"submodule.mem_map_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_hom.ker_square_lift (f : A →ₐ[R] B) : A ⧸ f.to_ring_hom.ker ^ 2 →ₐ[R] B | begin
refine { commutes' := _, ..(ideal.quotient.lift (f.to_ring_hom.ker ^ 2) f.to_ring_hom _) },
{ intros a ha, exact ideal.pow_le_self two_ne_zero ha },
{ intro r, rw [is_scalar_tower.algebra_map_apply R A, ring_hom.to_fun_eq_coe,
ideal.quotient.algebra_map_eq, ideal.quotient.lift_mk], exact f.map_algebra... | def | alg_hom.ker_square_lift | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"ideal.pow_le_self",
"ideal.quotient.algebra_map_eq",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"is_scalar_tower.algebra_map_apply",
"ring_hom.to_fun_eq_coe",
"two_ne_zero"
] | The lift of `f : A →ₐ[R] B` to `A ⧸ J ^ 2 →ₐ[R] B` with `J` being the kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.alg_hom.ker_ker_sqare_lift (f : A →ₐ[R] B) :
f.ker_square_lift.to_ring_hom.ker = f.to_ring_hom.ker.cotangent_ideal | begin
apply le_antisymm,
{ intros x hx, obtain ⟨x, rfl⟩ := ideal.quotient.mk_surjective x, exact ⟨x, hx, rfl⟩ },
{ rintros _ ⟨x, hx, rfl⟩, exact hx }
end | lemma | alg_hom.ker_ker_sqare_lift | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"ideal.quotient.mk_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_cotangent : ((R ⧸ I ^ 2) ⧸ I.cotangent_ideal) ≃+* R ⧸ I | begin
refine (ideal.quot_equiv_of_eq (ideal.map_eq_submodule_map _ _).symm).trans _,
refine (double_quot.quot_quot_equiv_quot_sup _ _).trans _,
exact (ideal.quot_equiv_of_eq (sup_eq_right.mpr $ ideal.pow_le_self two_ne_zero)),
end | def | ideal.quot_cotangent | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [
"double_quot.quot_quot_equiv_quot_sup",
"ideal.map_eq_submodule_map",
"ideal.pow_le_self",
"ideal.quot_equiv_of_eq",
"two_ne_zero"
] | The quotient ring of `I ⧸ I ^ 2` is `R ⧸ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotangent_space : Type* | (maximal_ideal R).cotangent | def | local_ring.cotangent_space | ring_theory.ideal | src/ring_theory/ideal/cotangent.lean | [
"ring_theory.ideal.operations",
"algebra.module.torsion",
"algebra.ring.idempotents",
"linear_algebra.finite_dimensional",
"ring_theory.ideal.local_ring"
] | [] | The `A ⧸ I`-vector space `I ⧸ I ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_elem_iff_of_fg {R : Type*} [comm_ring R] (I : ideal R)
(h : I.fg) :
is_idempotent_elem I ↔ ∃ e : R, is_idempotent_elem e ∧ I = R ∙ e | begin
split,
{ intro e,
obtain ⟨r, hr, hr'⟩ := submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h
(by { rw [smul_eq_mul], exact e.ge }),
simp_rw smul_eq_mul at hr',
refine ⟨r, hr' r hr, antisymm _ ((submodule.span_singleton_le_iff_mem _ _).mpr hr)⟩,
intros x hx,
rw ← hr' x hx,
... | lemma | ideal.is_idempotent_elem_iff_of_fg | ring_theory.ideal | src/ring_theory/ideal/idempotent_fg.lean | [
"algebra.ring.idempotents",
"ring_theory.finiteness"
] | [
"comm_ring",
"ideal",
"ideal.span_singleton_mul_span_singleton",
"is_idempotent_elem",
"mul_comm",
"smul_eq_mul",
"submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul",
"submodule.span_singleton_le_iff_mem"
] | A finitely generated idempotent ideal is generated by an idempotent element | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_elem_iff_eq_bot_or_top {R : Type*} [comm_ring R] [is_domain R]
(I : ideal R) (h : I.fg) :
is_idempotent_elem I ↔ I = ⊥ ∨ I = ⊤ | begin
split,
{ intro H,
obtain ⟨e, he, rfl⟩ := (I.is_idempotent_elem_iff_of_fg h).mp H,
simp only [ideal.submodule_span_eq, ideal.span_singleton_eq_bot],
apply or_of_or_of_imp_of_imp (is_idempotent_elem.iff_eq_zero_or_one.mp he) id,
rintro rfl,
simp },
{ rintro (rfl|rfl); simp [is_idempotent_e... | lemma | ideal.is_idempotent_elem_iff_eq_bot_or_top | ring_theory.ideal | src/ring_theory/ideal/idempotent_fg.lean | [
"algebra.ring.idempotents",
"ring_theory.finiteness"
] | [
"comm_ring",
"ideal",
"ideal.span_singleton_eq_bot",
"ideal.submodule_span_eq",
"is_domain",
"is_idempotent_elem",
"or_of_or_of_imp_of_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring (R : Type u) [semiring R] extends nontrivial R : Prop | of_is_unit_or_is_unit_of_add_one ::
(is_unit_or_is_unit_of_add_one {a b : R} (h : a + b = 1) : is_unit a ∨ is_unit b) | class | local_ring | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"nontrivial",
"semiring"
] | A semiring is local if it is nontrivial and `a` or `b` is a unit whenever `a + b = 1`.
Note that `local_ring` is a predicate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_unit_or_is_unit_of_is_unit_add [nontrivial R]
(h : ∀ a b : R, is_unit (a + b) → is_unit a ∨ is_unit b) :
local_ring R | ⟨λ a b hab, h a b $ hab.symm ▸ is_unit_one⟩ | lemma | local_ring.of_is_unit_or_is_unit_of_is_unit_add | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"local_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nonunits_add [nontrivial R]
(h : ∀ a b : R, a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R) :
local_ring R | ⟨λ a b hab, or_iff_not_and_not.2 $ λ H, h a b H.1 H.2 $ hab.symm ▸ is_unit_one⟩ | lemma | local_ring.of_nonunits_add | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"local_ring",
"nontrivial",
"nonunits"
] | A semiring is local if it is nontrivial and the set of nonunits is closed under the addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_unique_max_ideal (h : ∃! I : ideal R, I.is_maximal) :
local_ring R | @of_nonunits_add _ _ (nontrivial_of_ne (0 : R) 1 $
let ⟨I, Imax, _⟩ := h in (λ (H : 0 = 1), Imax.1.1 $ I.eq_top_iff_one.2 $ H ▸ I.zero_mem)) $
λ x y hx hy H,
let ⟨I, Imax, Iuniq⟩ := h in
let ⟨Ix, Ixmax, Hx⟩ := exists_max_ideal_of_mem_nonunits hx in
let ⟨Iy, Iymax, Hy⟩ := exists_max_ideal_of_mem_nonunits... | lemma | local_ring.of_unique_max_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"exists_max_ideal_of_mem_nonunits",
"ideal",
"local_ring",
"nontrivial_of_ne"
] | A semiring is local if it has a unique maximal ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_unique_nonzero_prime (h : ∃! P : ideal R, P ≠ ⊥ ∧ ideal.is_prime P) :
local_ring R | of_unique_max_ideal begin
rcases h with ⟨P, ⟨hPnonzero, hPnot_top, _⟩, hPunique⟩,
refine ⟨P, ⟨⟨hPnot_top, _⟩⟩, λ M hM, hPunique _ ⟨_, ideal.is_maximal.is_prime hM⟩⟩,
{ refine ideal.maximal_of_no_maximal (λ M hPM hM, ne_of_lt hPM _),
exact (hPunique _ ⟨ne_bot_of_gt hPM, ideal.is_maximal.is_prime hM⟩).symm },
... | lemma | local_ring.of_unique_nonzero_prime | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal",
"ideal.is_maximal.is_prime",
"ideal.is_prime",
"ideal.maximal_of_no_maximal",
"local_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_or_is_unit_of_is_unit_add {a b : R} (h : is_unit (a + b)) :
is_unit a ∨ is_unit b | begin
rcases h with ⟨u, hu⟩,
rw [←units.inv_mul_eq_one, mul_add] at hu,
apply or.imp _ _ (is_unit_or_is_unit_of_add_one hu);
exact is_unit_of_mul_is_unit_right,
end | lemma | local_ring.is_unit_or_is_unit_of_is_unit_add | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"is_unit_of_mul_is_unit_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonunits_add {a b : R} (ha : a ∈ nonunits R) (hb : b ∈ nonunits R) : a + b ∈ nonunits R | λ H, not_or ha hb (is_unit_or_is_unit_of_is_unit_add H) | lemma | local_ring.nonunits_add | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maximal_ideal : ideal R | { carrier := nonunits R,
zero_mem' := zero_mem_nonunits.2 $ zero_ne_one,
add_mem' := λ x y hx hy, nonunits_add hx hy,
smul_mem' := λ a x, mul_mem_nonunits_right } | def | local_ring.maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal",
"mul_mem_nonunits_right",
"nonunits",
"zero_ne_one"
] | The ideal of elements that are not units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_ideal.is_maximal : (maximal_ideal R).is_maximal | begin
rw ideal.is_maximal_iff,
split,
{ intro h, apply h, exact is_unit_one },
{ intros I x hI hx H,
erw not_not at hx,
rcases hx with ⟨u,rfl⟩,
simpa using I.mul_mem_left ↑u⁻¹ H }
end | instance | local_ring.maximal_ideal.is_maximal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.is_maximal_iff",
"is_unit_one",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maximal_ideal_unique : ∃! I : ideal R, I.is_maximal | ⟨maximal_ideal R, maximal_ideal.is_maximal R,
λ I hI, hI.eq_of_le (maximal_ideal.is_maximal R).1.1 $
λ x hx, hI.1.1 ∘ I.eq_top_of_is_unit_mem hx⟩ | lemma | local_ring.maximal_ideal_unique | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_maximal_ideal {I : ideal R} (hI : I.is_maximal) : I = maximal_ideal R | unique_of_exists_unique (maximal_ideal_unique R) hI $ maximal_ideal.is_maximal R | lemma | local_ring.eq_maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_maximal_ideal {J : ideal R} (hJ : J ≠ ⊤) : J ≤ maximal_ideal R | begin
rcases ideal.exists_le_maximal J hJ with ⟨M, hM1, hM2⟩,
rwa ←eq_maximal_ideal hM1
end | lemma | local_ring.le_maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal",
"ideal.exists_le_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_maximal_ideal (x) : x ∈ maximal_ideal R ↔ x ∈ nonunits R | iff.rfl | lemma | local_ring.mem_maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field_iff_maximal_ideal_eq :
is_field R ↔ maximal_ideal R = ⊥ | not_iff_not.mp ⟨ring.ne_bot_of_is_maximal_of_not_is_field infer_instance,
λ h, ring.not_is_field_iff_exists_prime.mpr ⟨_, h, ideal.is_maximal.is_prime' _⟩⟩ | lemma | local_ring.is_field_iff_maximal_ideal_eq | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.is_maximal.is_prime'",
"is_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_unit_or_is_unit_one_sub_self [nontrivial R]
(h : ∀ a : R, is_unit a ∨ is_unit (1 - a)) : local_ring R | ⟨λ a b hab, add_sub_cancel' a b ▸ hab.symm ▸ h a⟩ | lemma | local_ring.of_is_unit_or_is_unit_one_sub_self | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"local_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_or_is_unit_one_sub_self (a : R) : is_unit a ∨ is_unit (1 - a) | is_unit_or_is_unit_of_is_unit_add $ (add_sub_cancel'_right a 1).symm ▸ is_unit_one | lemma | local_ring.is_unit_or_is_unit_one_sub_self | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"is_unit_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_mem_nonunits_one_sub_self (a : R) (h : 1 - a ∈ nonunits R) :
is_unit a | or_iff_not_imp_right.1 (is_unit_or_is_unit_one_sub_self a) h | lemma | local_ring.is_unit_of_mem_nonunits_one_sub_self | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_one_sub_self_of_mem_nonunits (a : R) (h : a ∈ nonunits R) :
is_unit (1 - a) | or_iff_not_imp_left.1 (is_unit_or_is_unit_one_sub_self a) h | lemma | local_ring.is_unit_one_sub_self_of_mem_nonunits | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective' [comm_ring S] [nontrivial S] (f : R →+* S) (hf : function.surjective f) :
local_ring S | of_is_unit_or_is_unit_one_sub_self
begin
intros b,
obtain ⟨a, rfl⟩ := hf b,
apply (is_unit_or_is_unit_one_sub_self a).imp f.is_unit_map _,
rw [← f.map_one, ← f.map_sub],
apply f.is_unit_map,
end | lemma | local_ring.of_surjective' | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_ring",
"local_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_eq_maximal_ideal (I : ideal R) (h : I ≠ ⊤) :
I.jacobson = local_ring.maximal_ideal R | begin
apply le_antisymm,
{ exact Inf_le ⟨local_ring.le_maximal_ideal h, local_ring.maximal_ideal.is_maximal R⟩ },
{ exact le_Inf (λ J (hJ : I ≤ J ∧ J.is_maximal),
le_of_eq (local_ring.eq_maximal_ideal hJ.2).symm) }
end | lemma | local_ring.jacobson_eq_maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"Inf_le",
"ideal",
"le_Inf",
"local_ring.eq_maximal_ideal",
"local_ring.maximal_ideal",
"local_ring.maximal_ideal.is_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom [semiring R] [semiring S] (f : R →+* S) : Prop | (map_nonunit : ∀ a, is_unit (f a) → is_unit a) | class | is_local_ring_hom | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_unit",
"map_nonunit",
"semiring"
] | A local ring homomorphism is a homomorphism `f` between local rings such that `a` in the domain
is a unit if `f a` is a unit for any `a`. See `local_ring.local_hom_tfae` for other equivalent
definitions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_ring_hom_id (R : Type*) [semiring R] : is_local_ring_hom (ring_hom.id R) | { map_nonunit := λ a, id } | instance | is_local_ring_hom_id | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"map_nonunit",
"ring_hom.id",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_map_iff (f : R →+* S) [is_local_ring_hom f] (a) :
is_unit (f a) ↔ is_unit a | ⟨is_local_ring_hom.map_nonunit a, f.is_unit_map⟩ | lemma | is_unit_map_iff | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_nonunits_iff (f : R →+* S) [is_local_ring_hom f] (a) :
f a ∈ nonunits S ↔ a ∈ nonunits R | ⟨λ h ha, h $ (is_unit_map_iff f a).mpr ha, λ h ha, h $ (is_unit_map_iff f a).mp ha⟩ | lemma | map_mem_nonunits_iff | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"is_unit_map_iff",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_comp
(g : S →+* T) (f : R →+* S) [is_local_ring_hom g] [is_local_ring_hom f] :
is_local_ring_hom (g.comp f) | { map_nonunit := λ a, is_local_ring_hom.map_nonunit a ∘ is_local_ring_hom.map_nonunit (f a) } | instance | is_local_ring_hom_comp | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"map_nonunit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_equiv (f : R ≃+* S) :
is_local_ring_hom (f : R →+* S) | { map_nonunit := λ a ha,
begin
convert (f.symm : S →+* R).is_unit_map ha,
exact (ring_equiv.symm_apply_apply f a).symm,
end } | instance | is_local_ring_hom_equiv | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"map_nonunit",
"ring_equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_map_unit (f : R →+* S) [is_local_ring_hom f]
(a) (h : is_unit (f a)) : is_unit a | is_local_ring_hom.map_nonunit a h | lemma | is_unit_of_map_unit | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_irreducible_map (f : R →+* S) [h : is_local_ring_hom f] {x}
(hfx : irreducible (f x)) : irreducible x | ⟨λ h, hfx.not_unit $ is_unit.map f h, λ p q hx, let ⟨H⟩ := h in
or.imp (H p) (H q) $ hfx.is_unit_or_is_unit $ f.map_mul p q ▸ congr_arg f hx⟩ | theorem | of_irreducible_map | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"irreducible",
"is_local_ring_hom",
"is_unit.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_of_comp (f : R →+* S) (g : S →+* T) [is_local_ring_hom (g.comp f)] :
is_local_ring_hom f | ⟨λ a ha, (is_unit_map_iff (g.comp f) _).mp (g.is_unit_map ha)⟩ | lemma | is_local_ring_hom_of_comp | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"is_unit_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ring_hom.domain_local_ring {R S : Type*} [comm_semiring R] [comm_semiring S]
[H : _root_.local_ring S] (f : R →+* S)
[is_local_ring_hom f] : _root_.local_ring R | begin
haveI : nontrivial R := pullback_nonzero f f.map_zero f.map_one,
apply local_ring.of_nonunits_add,
intros a b,
simp_rw [←map_mem_nonunits_iff f, f.map_add],
exact local_ring.nonunits_add
end | lemma | ring_hom.domain_local_ring | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_semiring",
"is_local_ring_hom",
"local_ring.nonunits_add",
"local_ring.of_nonunits_add",
"nontrivial",
"pullback_nonzero"
] | If `f : R →+* S` is a local ring hom, then `R` is a local ring if `S` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nonunit (f : R →+* S) [is_local_ring_hom f] (a : R) (h : a ∈ maximal_ideal R) :
f a ∈ maximal_ideal S | λ H, h $ is_unit_of_map_unit f a H | lemma | map_nonunit | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom",
"is_unit_of_map_unit"
] | The image of the maximal ideal of the source is contained within the maximal ideal of the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_hom_tfae (f : R →+* S) :
tfae [is_local_ring_hom f,
f '' (maximal_ideal R).1 ⊆ maximal_ideal S,
(maximal_ideal R).map f ≤ maximal_ideal S,
maximal_ideal R ≤ (maximal_ideal S).comap f,
(maximal_ideal S).comap f = maximal_ideal R] | begin
tfae_have : 1 → 2, rintros _ _ ⟨a,ha,rfl⟩,
resetI, exact map_nonunit f a ha,
tfae_have : 2 → 4, exact set.image_subset_iff.1,
tfae_have : 3 ↔ 4, exact ideal.map_le_iff_le_comap,
tfae_have : 4 → 1, intro h, fsplit, exact λ x, not_imp_not.1 (@h x),
tfae_have : 1 → 5, intro, resetI, ext,
exact not_... | theorem | local_ring.local_hom_tfae | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.map_le_iff_le_comap",
"is_local_ring_hom",
"is_unit_map_iff",
"map_nonunit"
] | A ring homomorphism between local rings is a local ring hom iff it reflects units,
i.e. any preimage of a unit is still a unit. https://stacks.math.columbia.edu/tag/07BJ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_surjective [comm_semiring R] [local_ring R] [comm_semiring S] [nontrivial S]
(f : R →+* S) [is_local_ring_hom f] (hf : function.surjective f) :
local_ring S | of_is_unit_or_is_unit_of_is_unit_add
begin
intros a b hab,
obtain ⟨a, rfl⟩ := hf a,
obtain ⟨b, rfl⟩ := hf b,
rw ←map_add at hab,
exact (is_unit_or_is_unit_of_is_unit_add $ is_local_ring_hom.map_nonunit _ hab).imp
f.is_unit_map f.is_unit_map
end | lemma | local_ring.of_surjective | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_semiring",
"is_local_ring_hom",
"local_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_units_map_of_local_ring_hom [comm_ring R] [comm_ring S]
(f : R →+* S) (hf : function.surjective f) (h : is_local_ring_hom f) :
function.surjective (units.map $ f.to_monoid_hom) | begin
intro a,
obtain ⟨b,hb⟩ := hf (a : S),
use (is_unit_of_map_unit f _ (by { rw hb, exact units.is_unit _})).unit, ext, exact hb,
end | lemma | local_ring.surjective_units_map_of_local_ring_hom | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_ring",
"is_local_ring_hom",
"is_unit_of_map_unit",
"units.is_unit",
"units.map"
] | If `f : R →+* S` is a surjective local ring hom, then the induced units map is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residue_field | R ⧸ maximal_ideal R | def | local_ring.residue_field | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [] | The residue field of a local ring is the quotient of the ring by its maximal ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residue_field.field : field (residue_field R) | ideal.quotient.field (maximal_ideal R) | instance | local_ring.residue_field.field | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"field",
"ideal.quotient.field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
residue : R →+* (residue_field R) | ideal.quotient.mk _ | def | local_ring.residue | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.quotient.mk"
] | The quotient map from a local ring to its residue field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residue_field.algebra : algebra R (residue_field R) | ideal.quotient.algebra _ | instance | local_ring.residue_field.algebra | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"algebra",
"ideal.quotient.algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
residue_field.algebra_map_eq : algebra_map R (residue_field R) = residue R | rfl | lemma | local_ring.residue_field.algebra_map_eq | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift {R S : Type*} [comm_ring R] [local_ring R] [field S]
(f : R →+* S) [is_local_ring_hom f] : local_ring.residue_field R →+* S | ideal.quotient.lift _ f (λ a ha,
classical.by_contradiction (λ h, ha (is_unit_of_map_unit f a (is_unit_iff_ne_zero.mpr h)))) | def | local_ring.residue_field.lift | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_ring",
"field",
"ideal.quotient.lift",
"is_local_ring_hom",
"is_unit_of_map_unit",
"lift",
"local_ring",
"local_ring.residue_field"
] | A local ring homomorphism into a field can be descended onto the residue field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp_residue {R S : Type*} [comm_ring R] [local_ring R] [field S] (f : R →+* S)
[is_local_ring_hom f] : (lift f).comp (residue R) = f | ring_hom.ext (λ _, rfl) | lemma | local_ring.residue_field.lift_comp_residue | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_ring",
"field",
"is_local_ring_hom",
"lift",
"local_ring",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_residue_apply {R S : Type*} [comm_ring R] [local_ring R] [field S] (f : R →+* S)
[is_local_ring_hom f] (x) : lift f (residue R x) = f x | rfl | lemma | local_ring.residue_field.lift_residue_apply | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_ring",
"field",
"is_local_ring_hom",
"lift",
"local_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : R →+* S) [is_local_ring_hom f] : residue_field R →+* residue_field S | ideal.quotient.lift (maximal_ideal R) ((ideal.quotient.mk _).comp f) $
λ a ha,
begin
erw ideal.quotient.eq_zero_iff_mem,
exact map_nonunit f a ha
end | def | local_ring.residue_field.map | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.lift",
"ideal.quotient.mk",
"is_local_ring_hom",
"map_nonunit"
] | The map on residue fields induced by a local homomorphism between local rings | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_id :
local_ring.residue_field.map (ring_hom.id R) = ring_hom.id (local_ring.residue_field R) | ideal.quotient.ring_hom_ext $ ring_hom.ext $ λx, rfl | lemma | local_ring.residue_field.map_id | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.quotient.ring_hom_ext",
"local_ring.residue_field",
"local_ring.residue_field.map",
"map_id",
"ring_hom.ext",
"ring_hom.id"
] | Applying `residue_field.map` to the identity ring homomorphism gives the identity
ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp (f : T →+* R) (g : R →+* S) [is_local_ring_hom f] [is_local_ring_hom g] :
local_ring.residue_field.map (g.comp f) =
(local_ring.residue_field.map g).comp (local_ring.residue_field.map f) | ideal.quotient.ring_hom_ext $ ring_hom.ext $ λx, rfl | lemma | local_ring.residue_field.map_comp | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"ideal.quotient.ring_hom_ext",
"is_local_ring_hom",
"local_ring.residue_field.map",
"map_comp",
"ring_hom.ext"
] | The composite of two `residue_field.map`s is the `residue_field.map` of the composite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp_residue (f : R →+* S) [is_local_ring_hom f] :
(residue_field.map f).comp (residue R) = (residue S).comp f | rfl | lemma | local_ring.residue_field.map_comp_residue | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_residue (f : R →+* S) [is_local_ring_hom f] (r : R) :
residue_field.map f (residue R r) = residue S (f r) | rfl | lemma | local_ring.residue_field.map_residue | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"is_local_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_apply (x : residue_field R) : map (ring_hom.id R) x = x | fun_like.congr_fun map_id x | lemma | local_ring.residue_field.map_id_apply | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"fun_like.congr_fun",
"map_id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (f : R →+* S) (g : S →+* T) (x : residue_field R)
[is_local_ring_hom f] [is_local_ring_hom g] :
map g (map f x) = map (g.comp f) x | fun_like.congr_fun (map_comp f g).symm x | lemma | local_ring.residue_field.map_map | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"fun_like.congr_fun",
"is_local_ring_hom",
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv (f : R ≃+* S) : local_ring.residue_field R ≃+* local_ring.residue_field S | { to_fun := map (f : R →+* S),
inv_fun := map (f.symm : S →+* R),
left_inv := λ x, by simp only [map_map, ring_equiv.symm_comp, map_id, ring_hom.id_apply],
right_inv := λ x, by simp only [map_map, ring_equiv.comp_symm, map_id, ring_hom.id_apply],
map_mul' := ring_hom.map_mul _,
map_add' := ring_hom.map_add _ ... | def | local_ring.residue_field.map_equiv | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"inv_fun",
"local_ring.residue_field",
"map_id",
"ring_equiv.comp_symm",
"ring_equiv.symm_comp",
"ring_hom.id_apply",
"ring_hom.map_add",
"ring_hom.map_mul"
] | A ring isomorphism defines an isomorphism of residue fields. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_equiv.symm (f : R ≃+* S) : (map_equiv f).symm = map_equiv f.symm | rfl | lemma | local_ring.residue_field.map_equiv.symm | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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