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 |
|---|---|---|---|---|---|---|---|---|---|---|
map_pow (n : ℕ) : map f (I^n) = (map f I)^n | map_pow (map_hom f) I n | theorem | ideal.map_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_radical : comap f (radical K) = radical (comap f K) | by { ext, simpa only [radical, mem_comap, map_pow] } | theorem | ideal.comap_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical.comap (hK : K.is_radical) : (comap f K).is_radical | by { rw [←hK.radical, comap_radical], apply radical_is_radical } | theorem | ideal.is_radical.comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_radical"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_radical_le : map f (radical I) ≤ radical (map f I) | map_le_iff_le_comap.2 $ λ r ⟨n, hrni⟩, ⟨n, map_pow f r n ▸ mem_map_of_mem f hrni⟩ | theorem | ideal.map_radical_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_mul : comap f K * comap f L ≤ comap f (K * L) | map_le_iff_le_comap.1 $ (map_mul f (comap f K) (comap f L)).symm ▸
mul_mono (map_le_iff_le_comap.2 $ le_rfl) (map_le_iff_le_comap.2 $ le_rfl) | theorem | ideal.le_comap_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"le_rfl",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_pow (n : ℕ) :
(K.comap f) ^ n ≤ (K ^ n).comap f | begin
induction n,
{ rw [pow_zero, pow_zero, ideal.one_eq_top, ideal.one_eq_top], exact rfl.le },
{ rw [pow_succ, pow_succ], exact (ideal.mul_mono_right n_ih).trans (ideal.le_comap_mul f) }
end | lemma | ideal.le_comap_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.le_comap_mul",
"ideal.mul_mono_right",
"ideal.one_eq_top",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primary (I : ideal R) : Prop | I ≠ ⊤ ∧ ∀ {x y : R}, x * y ∈ I → x ∈ I ∨ y ∈ radical I | def | ideal.is_primary | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | A proper ideal `I` is primary iff `xy ∈ I` implies `x ∈ I` or `y ∈ radical I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime.is_primary {I : ideal R} (hi : is_prime I) : is_primary I | ⟨hi.1, λ x y hxy, (hi.mem_or_mem hxy).imp id $ λ hyi, le_radical hyi⟩ | theorem | ideal.is_prime.is_primary | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_radical_of_pow_mem {I : ideal R} {x : R} {m : ℕ} (hx : x ^ m ∈ radical I) :
x ∈ radical I | radical_idem I ▸ ⟨m, hx⟩ | theorem | ideal.mem_radical_of_pow_mem | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_radical {I : ideal R} (hi : is_primary I) : is_prime (radical I) | ⟨mt radical_eq_top.1 hi.1, λ x y ⟨m, hxy⟩, begin
rw mul_pow at hxy, cases hi.2 hxy,
{ exact or.inl ⟨m, h⟩ },
{ exact or.inr (mem_radical_of_pow_mem h) }
end⟩ | theorem | ideal.is_prime_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"mul_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primary_inf {I J : ideal R} (hi : is_primary I) (hj : is_primary J)
(hij : radical I = radical J) : is_primary (I ⊓ J) | ⟨ne_of_lt $ lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1), λ x y ⟨hxyi, hxyj⟩,
begin
rw [radical_inf, hij, inf_idem],
cases hi.2 hxyi with hxi hyi, cases hj.2 hxyj with hxj hyj,
{ exact or.inl ⟨hxi, hxj⟩ },
{ exact or.inr hyj },
{ rw hij at hyi, exact or.inr hyi }
end⟩ | theorem | ideal.is_primary_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inf_idem",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsupp_total : (ι →₀ I) →ₗ[R] M | (finsupp.total ι M R v).comp (finsupp.map_range.linear_map I.subtype) | def | ideal.finsupp_total | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finsupp.map_range.linear_map",
"finsupp.total"
] | A variant of `finsupp.total` that takes in vectors valued in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finsupp_total_apply (f : ι →₀ I) :
finsupp_total ι M I v f = f.sum (λ i x, (x : R) • v i) | begin
dsimp [finsupp_total],
rw [finsupp.total_apply, finsupp.sum_map_range_index],
exact λ _, zero_smul _ _
end | lemma | ideal.finsupp_total_apply | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finsupp.total_apply",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finsupp_total_apply_eq_of_fintype [fintype ι] (f : ι →₀ I) :
finsupp_total ι M I v f = ∑ i, (f i : R) • v i | by { rw [finsupp_total_apply, finsupp.sum_fintype], exact λ _, zero_smul _ _ } | lemma | ideal.finsupp_total_apply_eq_of_fintype | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"fintype",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_finsupp_total :
(finsupp_total ι M I v).range = I • (submodule.span R (set.range v)) | begin
ext,
rw submodule.mem_ideal_smul_span_iff_exists_sum,
refine ⟨λ ⟨f, h⟩, ⟨finsupp.map_range.linear_map I.subtype f, λ i, (f i).2, h⟩, _⟩,
rintro ⟨a, ha, rfl⟩,
classical,
refine ⟨a.map_range (λ r, if h : r ∈ I then ⟨r, h⟩ else 0) (by split_ifs; refl), _⟩,
rw [finsupp_total_apply, finsupp.sum_map_range... | lemma | ideal.range_finsupp_total | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.range",
"submodule.mem_ideal_smul_span_iff_exists_sum",
"submodule.span",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_span_singleton (b : basis ι R S) {x : S} (hx : x ≠ 0) :
basis ι R (span ({x} : set S)) | b.map $ ((linear_equiv.of_injective (algebra.lmul R S x) (linear_map.mul_injective hx)) ≪≫ₗ
(linear_equiv.of_eq _ _ (by { ext, simp [mem_span_singleton', mul_comm] })) ≪≫ₗ
((submodule.restrict_scalars_equiv R S S (ideal.span ({x} : set S))).restrict_scalars R)) | def | ideal.basis_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra.lmul",
"basis",
"ideal.span",
"linear_equiv.of_eq",
"linear_equiv.of_injective",
"linear_map.mul_injective",
"mul_comm",
"restrict_scalars",
"submodule.restrict_scalars_equiv"
] | A basis on `S` gives a basis on `ideal.span {x}`, by multiplying everything by `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_span_singleton_apply (b : basis ι R S) {x : S} (hx : x ≠ 0) (i : ι) :
(basis_span_singleton b hx i : S) = x * b i | begin
simp only [basis_span_singleton, basis.map_apply, linear_equiv.trans_apply,
submodule.restrict_scalars_equiv_apply, linear_equiv.of_injective_apply,
linear_equiv.coe_of_eq_apply, linear_equiv.restrict_scalars_apply,
algebra.coe_lmul_eq_mul, linear_map.mul_apply']
end | lemma | ideal.basis_span_singleton_apply | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra.coe_lmul_eq_mul",
"basis",
"basis.map_apply",
"linear_equiv.coe_of_eq_apply",
"linear_equiv.of_injective_apply",
"linear_equiv.trans_apply",
"linear_map.mul_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constr_basis_span_singleton
{N : Type*} [semiring N] [module N S] [smul_comm_class R N S]
(b : basis ι R S) {x : S} (hx : x ≠ 0) :
b.constr N (coe ∘ basis_span_singleton b hx) = algebra.lmul R S x | b.ext (λ i, by erw [basis.constr_basis, function.comp_app, basis_span_singleton_apply,
linear_map.mul_apply']) | lemma | ideal.constr_basis_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra.lmul",
"basis",
"basis.constr_basis",
"linear_map.mul_apply'",
"module",
"semiring",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates.mk_ne_zero' {R : Type*} [comm_semiring R] {r : R} :
(associates.mk (ideal.span {r} : ideal R)) ≠ 0 ↔ (r ≠ 0) | by rw [associates.mk_ne_zero, ideal.zero_eq_bot, ne.def, ideal.span_singleton_eq_bot] | lemma | associates.mk_ne_zero' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"associates.mk",
"associates.mk_ne_zero",
"comm_semiring",
"ideal",
"ideal.span",
"ideal.span_singleton_eq_bot",
"ideal.zero_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.mem_ideal_iff {ι R S : Type*} [comm_ring R] [comm_ring S] [algebra R S]
{I : ideal S} (b : basis ι R I) {x : S} :
x ∈ I ↔ ∃ (c : ι →₀ R), x = finsupp.sum c (λ i x, x • b i) | (b.map ((I.restrict_scalars_equiv R _ _).restrict_scalars R).symm).mem_submodule_iff | lemma | basis.mem_ideal_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra",
"basis",
"comm_ring",
"ideal",
"restrict_scalars"
] | If `I : ideal S` has a basis over `R`,
`x ∈ I` iff it is a linear combination of basis vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.mem_ideal_iff' {ι R S : Type*} [fintype ι] [comm_ring R] [comm_ring S] [algebra R S]
{I : ideal S} (b : basis ι R I) {x : S} :
x ∈ I ↔ ∃ (c : ι → R), x = ∑ i, c i • b i | (b.map ((I.restrict_scalars_equiv R _ _).restrict_scalars R).symm).mem_submodule_iff' | lemma | basis.mem_ideal_iff' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra",
"basis",
"comm_ring",
"fintype",
"ideal",
"restrict_scalars"
] | If `I : ideal S` has a finite basis over `R`,
`x ∈ I` iff it is a linear combination of basis vectors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker : ideal R | ideal.comap f ⊥ | def | ring_hom.ker | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.comap"
] | Kernel of a ring homomorphism as an ideal of the domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ker {r} : r ∈ ker f ↔ f r = 0 | by rw [ker, ideal.mem_comap, submodule.mem_bot] | lemma | ring_hom.mem_ker | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mem_comap",
"submodule.mem_bot"
] | An element is in the kernel if and only if it maps to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_eq : ((ker f) : set R) = set.preimage f {0} | rfl | lemma | ring_hom.ker_eq | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_comap_bot (f : F) : ker f = ideal.comap f ⊥ | rfl | lemma | ring_hom.ker_eq_comap_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ker (f : S →+* R) (g : T →+* S) : f.ker.comap g = (f.comp g).ker | by rw [ring_hom.ker_eq_comap_bot, ideal.comap_comap, ring_hom.ker_eq_comap_bot] | lemma | ring_hom.comap_ker | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.comap_comap",
"ring_hom.ker_eq_comap_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_one_mem_ker [nontrivial S] (f : F) : (1:R) ∉ ker f | by { rw [mem_ker, map_one], exact one_ne_zero } | lemma | ring_hom.not_one_mem_ker | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_one",
"nontrivial",
"one_ne_zero"
] | If the target is not the zero ring, then one is not in the kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_ne_top [nontrivial S] (f : F) : ker f ≠ ⊤ | (ideal.ne_top_iff_one _).mpr $ not_one_mem_ker f | lemma | ring_hom.ker_ne_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.ne_top_iff_one",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_iff_ker_eq_bot : function.injective f ↔ ker f = ⊥ | by { rw [set_like.ext'_iff, ker_eq, set.ext_iff], exact injective_iff_map_eq_zero' f } | lemma | ring_hom.injective_iff_ker_eq_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.ext_iff",
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 | by { rw [← injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] } | lemma | ring_hom.ker_eq_bot_iff_eq_zero | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_coe_equiv (f : R ≃+* S) :
ker (f : R →+* S) = ⊥ | by simpa only [←injective_iff_ker_eq_bot] using equiv_like.injective f | lemma | ring_hom.ker_coe_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"equiv_like.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_equiv {F' : Type*} [ring_equiv_class F' R S] (f : F') :
ker f = ⊥ | by simpa only [←injective_iff_ker_eq_bot] using equiv_like.injective f | lemma | ring_hom.ker_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"equiv_like.injective",
"ring_equiv_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_is_prime {F : Type*} [ring R] [ring S] [is_domain S] [ring_hom_class F R S]
(f : F) : (ker f).is_prime | ⟨by { rw [ne.def, ideal.eq_top_iff_one], exact not_one_mem_ker f },
λ x y, by simpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _⟩ | lemma | ring_hom.ker_is_prime | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.eq_top_iff_one",
"is_domain",
"map_mul",
"ring",
"ring_hom_class"
] | The kernel of a homomorphism to a domain is a prime ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_is_maximal_of_surjective {R K F : Type*} [ring R] [field K] [ring_hom_class F R K]
(f : F) (hf : function.surjective f) :
(ker f).is_maximal | begin
refine ideal.is_maximal_iff.mpr
⟨λ h1, one_ne_zero' K $ map_one f ▸ (mem_ker f).mp h1,
λ J x hJ hxf hxJ, _⟩,
obtain ⟨y, hy⟩ := hf (f x)⁻¹,
have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm,
rw H,
refine J.sub_mem (J.mul_mem_left _ hxJ) (hJ _),
rw mem_ker,
simp only [hy, map_sub, ... | lemma | ring_hom.ker_is_maximal_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"field",
"inv_mul_cancel",
"map_mul",
"map_one",
"one_ne_zero'",
"ring",
"ring_hom_class"
] | The kernel of a homomorphism to a field is a maximal ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_bot_iff_le_ker {I : ideal R} (f : F) : I.map f = ⊥ ↔ I ≤ (ring_hom.ker f) | by rw [ring_hom.ker, eq_bot_iff, map_le_iff_le_comap] | lemma | ideal.map_eq_bot_iff_le_ker | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"eq_bot_iff",
"ideal",
"ring_hom.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_le_comap {K : ideal S} (f : F) : ring_hom.ker f ≤ comap f K | λ x hx, mem_comap.2 (((ring_hom.mem_ker f).1 hx).symm ▸ K.zero_mem) | lemma | ideal.ker_le_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_hom.ker",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Inf {A : set (ideal R)} {f : F} (hf : function.surjective f) :
(∀ J ∈ A, ring_hom.ker f ≤ J) → map f (Inf A) = Inf (map f '' A) | begin
refine λ h, le_antisymm (le_Inf _) _,
{ intros j hj y hy,
cases (mem_map_iff_of_surjective f hf).1 hy with x hx,
cases (set.mem_image _ _ _).mp hj with J hJ,
rw [← hJ.right, ← hx.right],
exact mem_map_of_mem f (Inf_le_of_le hJ.left (le_of_eq rfl) hx.left) },
{ intros y hy,
cases hf y wit... | lemma | ideal.map_Inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"Inf_le_of_le",
"ideal",
"le_Inf",
"ring_hom.ker",
"ring_hom.mem_ker",
"set.mem_image",
"submodule.mem_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_prime_of_surjective {f : F} (hf : function.surjective f) {I : ideal R}
[H : is_prime I] (hk : ring_hom.ker f ≤ I) : is_prime (map f I) | begin
refine ⟨λ h, H.ne_top (eq_top_iff.2 _), λ x y, _⟩,
{ replace h := congr_arg (comap f) h,
rw [comap_map_of_surjective _ hf, comap_top] at h,
exact h ▸ sup_le (le_of_eq rfl) hk },
{ refine λ hxy, (hf x).rec_on (λ a ha, (hf y).rec_on (λ b hb, _)),
rw [← ha, ← hb, ← _root_.map_mul f, mem_map_iff_of_... | theorem | ideal.map_is_prime_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_hom.ker",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_bot_iff_of_injective {I : ideal R} {f : F} (hf : function.injective f) :
I.map f = ⊥ ↔ I = ⊥ | by rw [map_eq_bot_iff_le_ker, (ring_hom.injective_iff_ker_eq_bot f).mp hf, le_bot_iff] | lemma | ideal.map_eq_bot_iff_of_injective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"le_bot_iff",
"ring_hom.injective_iff_ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_prime_of_equiv {F' : Type*} [ring_equiv_class F' R S]
(f : F') {I : ideal R} [is_prime I] :
is_prime (map f I) | map_is_prime_of_surjective (equiv_like.surjective f) $ by simp only [ring_hom.ker_equiv, bot_le] | theorem | ideal.map_is_prime_of_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"bot_le",
"equiv_like.surjective",
"ideal",
"ring_equiv_class",
"ring_hom.ker_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_iff_sup_ker_eq_of_surjective {I J : ideal R} (f : R →+* S)
(hf : function.surjective f) : map f I = map f J ↔ I ⊔ f.ker = J ⊔ f.ker | by rw [← (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf,
comap_map_of_surjective f hf, ring_hom.ker_eq_comap_bot] | theorem | ideal.map_eq_iff_sup_ker_eq_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_hom.ker_eq_comap_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_radical_of_surjective {f : R →+* S} (hf : function.surjective f) {I : ideal R}
(h : ring_hom.ker f ≤ I) : map f (I.radical) = (map f I).radical | begin
rw [radical_eq_Inf, radical_eq_Inf],
have : ∀ J ∈ {J : ideal R | I ≤ J ∧ J.is_prime}, f.ker ≤ J := λ J hJ, le_trans h hJ.left,
convert map_Inf hf this,
refine funext (λ j, propext ⟨_, _⟩),
{ rintros ⟨hj, hj'⟩,
haveI : j.is_prime := hj',
exact ⟨comap f j, ⟨⟨map_le_iff_le_comap.1 hj, comap_is_prim... | theorem | ideal.map_radical_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_hom.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_submodule : module (ideal R) (submodule R M) | { smul_add := smul_sup,
add_smul := sup_smul,
mul_smul := submodule.smul_assoc,
one_smul := by simp,
zero_smul := bot_smul,
smul_zero := smul_bot } | instance | submodule.module_submodule | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"add_smul",
"ideal",
"module",
"one_smul",
"smul_add",
"smul_zero",
"submodule",
"submodule.smul_assoc",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_right_inverse_aux
(hf : function.right_inverse f_inv f) (g : A →+* C) (hg : f.ker ≤ g.ker) :
B →+* C | { to_fun := λ b, g (f_inv b),
map_one' :=
begin
rw [← g.map_one, ← sub_eq_zero, ← g.map_sub, ← g.mem_ker],
apply hg,
rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_one],
exact hf 1
end,
map_mul' :=
begin
intros x y,
rw [← g.map_mul, ← sub_eq_zero, ← g.map_sub, ← g.mem_ker],
apply hg,... | def | ring_hom.lift_of_right_inverse_aux | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | Auxiliary definition used to define `lift_of_right_inverse` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of_right_inverse_aux_comp_apply
(hf : function.right_inverse f_inv f) (g : A →+* C) (hg : f.ker ≤ g.ker) (a : A) :
(f.lift_of_right_inverse_aux f_inv hf g hg) (f a) = g a | f.to_add_monoid_hom.lift_of_right_inverse_comp_apply f_inv hf ⟨g.to_add_monoid_hom, hg⟩ a | lemma | ring_hom.lift_of_right_inverse_aux_comp_apply | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_right_inverse
(hf : function.right_inverse f_inv f) : {g : A →+* C // f.ker ≤ g.ker} ≃ (B →+* C) | { to_fun := λ g, f.lift_of_right_inverse_aux f_inv hf g.1 g.2,
inv_fun := λ φ, ⟨φ.comp f, λ x hx, (mem_ker _).mpr $ by simp [(mem_ker _).mp hx]⟩,
left_inv := λ g, by
{ ext,
simp only [comp_apply, lift_of_right_inverse_aux_comp_apply, subtype.coe_mk,
subtype.val_eq_coe], },
right_inv := λ φ, by
{ ext... | def | ring_hom.lift_of_right_inverse | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"inv_fun",
"subtype.coe_mk",
"subtype.val_eq_coe"
] | `lift_of_right_inverse f hf g hg` is the unique ring homomorphism `φ`
* such that `φ.comp f = g` (`ring_hom.lift_of_right_inverse_comp`),
* where `f : A →+* B` is has a right_inverse `f_inv` (`hf`),
* and `g : B →+* C` satisfies `hg : f.ker ≤ g.ker`.
See `ring_hom.eq_lift_of_right_inverse` for the uniqueness lemma.
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of_surjective
(hf : function.surjective f) : {g : A →+* C // f.ker ≤ g.ker} ≃ (B →+* C) | f.lift_of_right_inverse (function.surj_inv hf) (function.right_inverse_surj_inv hf) | abbreviation | ring_hom.lift_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | A non-computable version of `ring_hom.lift_of_right_inverse` for when no computable right
inverse is available, that uses `function.surj_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of_right_inverse_comp_apply
(hf : function.right_inverse f_inv f) (g : {g : A →+* C // f.ker ≤ g.ker}) (x : A) :
(f.lift_of_right_inverse f_inv hf g) (f x) = g x | f.lift_of_right_inverse_aux_comp_apply f_inv hf g.1 g.2 x | lemma | ring_hom.lift_of_right_inverse_comp_apply | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_right_inverse_comp (hf : function.right_inverse f_inv f)
(g : {g : A →+* C // f.ker ≤ g.ker}) :
(f.lift_of_right_inverse f_inv hf g).comp f = g | ring_hom.ext $ f.lift_of_right_inverse_comp_apply f_inv hf g | lemma | ring_hom.lift_of_right_inverse_comp | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lift_of_right_inverse (hf : function.right_inverse f_inv f) (g : A →+* C)
(hg : f.ker ≤ g.ker) (h : B →+* C) (hh : h.comp f = g) :
h = (f.lift_of_right_inverse f_inv hf ⟨g, hg⟩) | begin
simp_rw ←hh,
exact ((f.lift_of_right_inverse f_inv hf).apply_symm_apply _).symm,
end | lemma | ring_hom.eq_lift_of_right_inverse | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_mem_comap_of_root_mem_of_eval_mem {r : S} (hr : r ∈ I) {p : R[X]}
(hp : p.eval₂ f r ∈ I) : p.coeff 0 ∈ I.comap f | begin
rw [←p.div_X_mul_X_add, eval₂_add, eval₂_C, eval₂_mul, eval₂_X] at hp,
refine mem_comap.mpr ((I.add_mem_iff_right _).mp hp),
exact I.mul_mem_left _ hr
end | lemma | ideal.coeff_zero_mem_comap_of_root_mem_of_eval_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_mem_comap_of_root_mem {r : S} (hr : r ∈ I) {p : R[X]}
(hp : p.eval₂ f r = 0) : p.coeff 0 ∈ I.comap f | coeff_zero_mem_comap_of_root_mem_of_eval_mem hr (hp.symm ▸ I.zero_mem) | lemma | ideal.coeff_zero_mem_comap_of_root_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem {r : S}
(r_non_zero_divisor : ∀ {x}, x * r = 0 → x = 0) (hr : r ∈ I)
{p : R[X]} : ∀ (p_ne_zero : p ≠ 0) (hp : p.eval₂ f r = 0),
∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ I.comap f | begin
refine p.rec_on_horner _ _ _,
{ intro h, contradiction },
{ intros p a coeff_eq_zero a_ne_zero ih p_ne_zero hp,
refine ⟨0, _, coeff_zero_mem_comap_of_root_mem hr hp⟩,
simp [coeff_eq_zero, a_ne_zero] },
{ intros p p_nonzero ih mul_nonzero hp,
rw [eval₂_mul, eval₂_X] at hp,
obtain ⟨i, hi, me... | lemma | ideal.exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_quotient_le_comap_map (P : ideal R[X]) :
function.injective ((map (map_ring_hom (quotient.mk (P.comap (C : R →+* R[X])))) P).quotient_map
(map_ring_hom (quotient.mk (P.comap (C : R →+* R[X])))) le_comap_map) | begin
refine quotient_map_injective' (le_of_eq _),
rw comap_map_of_surjective
(map_ring_hom (quotient.mk (P.comap (C : R →+* R[X]))))
(map_surjective (quotient.mk (P.comap (C : R →+* R[X]))) quotient.mk_surjective),
refine le_antisymm (sup_le le_rfl _) (le_sup_of_le_left le_rfl),
refine λ p hp, polyno... | lemma | ideal.injective_quotient_le_comap_map | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ideal",
"le_rfl",
"le_sup_of_le_left",
"quotient_map",
"sup_le"
] | Let `P` be an ideal in `R[x]`. The map
`R[x]/P → (R / (P ∩ R))[x] / (P / (P ∩ R))`
is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_mk_maps_eq (P : ideal R[X]) :
((quotient.mk (map (map_ring_hom (quotient.mk (P.comap (C : R →+* R[X])))) P)).comp C).comp
(quotient.mk (P.comap (C : R →+* R[X]))) =
((map (map_ring_hom (quotient.mk (P.comap (C : R →+* R[X])))) P).quotient_map
(map_ring_hom (quotient.mk (P.comap (C : R →+* R[X])))) ... | begin
refine ring_hom.ext (λ x, _),
repeat { rw [ring_hom.coe_comp, function.comp_app] },
rw [quotient_map_mk, coe_map_ring_hom, map_C],
end | lemma | ideal.quotient_mk_maps_eq | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ideal",
"quotient_map",
"ring_hom.coe_comp",
"ring_hom.ext"
] | The identity in this lemma asserts that the "obvious" square
```
R → (R / (P ∩ R))
↓ ↓
R[x] / P → (R / (P ∩ R))[x] / (P / (P ∩ R))
```
commutes. It is used, for instance, in the proof of `quotient_mk_comp_C_is_integral_of_jacobson`,
in the file `ring_theory/jacobson`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nonzero_mem_of_ne_bot {P : ideal R[X]}
(Pb : P ≠ ⊥) (hP : ∀ (x : R), C x ∈ P → x = 0) :
∃ p : R[X], p ∈ P ∧ (polynomial.map (quotient.mk (P.comap (C : R →+* R[X]))) p) ≠ 0 | begin
obtain ⟨m, hm⟩ := submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb),
refine ⟨m, submodule.coe_mem m, λ pp0, hm (submodule.coe_eq_zero.mp _)⟩,
refine (injective_iff_map_eq_zero
(polynomial.map_ring_hom (quotient.mk (P.comap (C : R →+* R[X]))))).mp _ _ pp0,
refine map_injective _ ((quotient.mk ... | lemma | ideal.exists_nonzero_mem_of_ne_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ideal",
"polynomial.map",
"polynomial.map_ring_hom",
"submodule.coe_mem",
"submodule.eq_bot_iff",
"submodule.nonzero_mem_of_bot_lt"
] | This technical lemma asserts the existence of a polynomial `p` in an ideal `P ⊂ R[x]`
that is non-zero in the quotient `R / (P ∩ R) [x]`. The assumptions are equivalent to
`P ≠ 0` and `P ∩ R = (0)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq_of_scalar_tower_quotient [algebra R S] [algebra (R ⧸ p) (S ⧸ P)]
[is_scalar_tower R (R ⧸ p) (S ⧸ P)]
(h : function.injective (algebra_map (R ⧸ p) (S ⧸ P))) :
comap (algebra_map R S) P = p | begin
ext x, split; rw [mem_comap, ← quotient.eq_zero_iff_mem, ← quotient.eq_zero_iff_mem,
quotient.mk_algebra_map, is_scalar_tower.algebra_map_apply _ (R ⧸ p),
quotient.algebra_map_eq],
{ intro hx,
exact (injective_iff_map_eq_zero (algebra_map (R ⧸ p) (S ⧸ P))).mp h _ hx },
{ intro hx,
rw [hx, ri... | lemma | ideal.comap_eq_of_scalar_tower_quotient | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra_map",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"ring_hom.map_zero"
] | If there is an injective map `R/p → S/P` such that following diagram commutes:
```
R → S
↓ ↓
R/p → S/P
```
then `P` lies over `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.algebra_quotient_of_le_comap (h : p ≤ comap f P) :
algebra (R ⧸ p) (S ⧸ P) | ring_hom.to_algebra $ quotient_map _ f h | def | ideal.quotient.algebra_quotient_of_le_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"quotient_map",
"ring_hom.to_algebra"
] | If `P` lies over `p`, then `R / p` has a canonical map to `S / P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.algebra_quotient_map_quotient :
algebra (R ⧸ p) (S ⧸ map f p) | by exact quotient.algebra_quotient_of_le_comap le_comap_map | instance | ideal.quotient.algebra_quotient_map_quotient | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra"
] | `R / p` has a canonical map to `S / pS`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.algebra_map_quotient_map_quotient (x : R) :
algebra_map (R ⧸ p) (S ⧸ map f p) (quotient.mk p x) = quotient.mk _ (f x) | rfl | lemma | ideal.quotient.algebra_map_quotient_map_quotient | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mk_smul_mk_quotient_map_quotient (x : R) (y : S) :
quotient.mk p x • quotient.mk (map f p) y = quotient.mk _ (f x * y) | rfl | lemma | ideal.quotient.mk_smul_mk_quotient_map_quotient | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.tower_quotient_map_quotient [algebra R S] :
is_scalar_tower R (R ⧸ p) (S ⧸ map (algebra_map R S) p) | is_scalar_tower.of_algebra_map_eq $ λ x,
by rw [quotient.algebra_map_eq, quotient.algebra_map_quotient_map_quotient,
quotient.mk_algebra_map] | instance | ideal.quotient.tower_quotient_map_quotient | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra_map",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_quotient.is_noetherian [algebra R S] [is_noetherian R S] (I : ideal R) :
is_noetherian (R ⧸ I) (S ⧸ ideal.map (algebra_map R S) I) | is_noetherian_of_tower R $
is_noetherian_of_surjective S (ideal.quotient.mkₐ R _).to_linear_map $
linear_map.range_eq_top.mpr ideal.quotient.mk_surjective | instance | ideal.quotient_map_quotient.is_noetherian | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra_map",
"ideal",
"ideal.map",
"ideal.quotient.mk_surjective",
"ideal.quotient.mkₐ",
"is_noetherian",
"is_noetherian_of_surjective",
"is_noetherian_of_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_coeff_ne_zero_mem_comap_of_root_mem
[is_domain S] {r : S} (r_ne_zero : r ≠ 0) (hr : r ∈ I)
{p : R[X]} : ∀ (p_ne_zero : p ≠ 0) (hp : p.eval₂ f r = 0),
∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ I.comap f | exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem
(λ _ h, or.resolve_right (mul_eq_zero.mp h) r_ne_zero) hr | lemma | ideal.exists_coeff_ne_zero_mem_comap_of_root_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff
[is_prime I] (hIJ : I ≤ J) {r : S} (hr : r ∈ (J : set S) \ I)
{p : R[X]} (p_ne_zero : p.map (quotient.mk (I.comap f)) ≠ 0) (hpI : p.eval₂ f r ∈ I) :
∃ i, p.coeff i ∈ (J.comap f : set R) \ (I.comap f) | begin
obtain ⟨hrJ, hrI⟩ := hr,
have rbar_ne_zero : quotient.mk I r ≠ 0 := mt (quotient.mk_eq_zero I).mp hrI,
have rbar_mem_J : quotient.mk I r ∈ J.map (quotient.mk I) := mem_map_of_mem _ hrJ,
have quotient_f : ∀ x ∈ I.comap f, (quotient.mk I).comp f x = 0,
{ simp [quotient.eq_zero_iff_mem] },
have rbar_root... | lemma | ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lt_comap_of_root_mem_sdiff [I.is_prime] (hIJ : I ≤ J)
{r : S} (hr : r ∈ (J : set S) \ I)
{p : R[X]} (p_ne_zero : p.map (quotient.mk (I.comap f)) ≠ 0) (hp : p.eval₂ f r ∈ I) :
I.comap f < J.comap f | let ⟨i, hJ, hI⟩ := exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff hIJ hr p_ne_zero hp
in set_like.lt_iff_le_and_exists.mpr ⟨comap_mono hIJ, p.coeff i, hJ, hI⟩ | lemma | ideal.comap_lt_comap_of_root_mem_sdiff | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_one_mem (h : (1 : S) ∈ I) (x) : x ∈ I | (I.eq_top_iff_one.mpr h).symm ▸ mem_top | lemma | ideal.mem_of_one_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_lt_comap_of_integral_mem_sdiff [algebra R S] [hI : I.is_prime] (hIJ : I ≤ J)
{x : S} (mem : x ∈ (J : set S) \ I) (integral : is_integral R x) :
I.comap (algebra_map R S) < J.comap (algebra_map R S) | begin
obtain ⟨p, p_monic, hpx⟩ := integral,
refine comap_lt_comap_of_root_mem_sdiff hIJ mem _ _,
swap,
{ apply map_monic_ne_zero p_monic,
apply quotient.nontrivial,
apply mt comap_eq_top_iff.mp,
apply hI.1 },
convert I.zero_mem
end | lemma | ideal.comap_lt_comap_of_integral_mem_sdiff | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra_map",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_of_root_mem [is_domain S] {r : S} (r_ne_zero : r ≠ 0) (hr : r ∈ I)
{p : R[X]} (p_ne_zero : p ≠ 0) (hp : p.eval₂ f r = 0) :
I.comap f ≠ ⊥ | λ h, let ⟨i, hi, mem⟩ := exists_coeff_ne_zero_mem_comap_of_root_mem r_ne_zero hr p_ne_zero hp in
absurd (mem_bot.mp (eq_bot_iff.mp h mem)) hi | lemma | ideal.comap_ne_bot_of_root_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_of_is_integral_of_is_maximal_comap
[algebra R S] (hRS : algebra.is_integral R S) (I : ideal S) [I.is_prime]
(hI : is_maximal (I.comap (algebra_map R S))) : is_maximal I | ⟨⟨mt comap_eq_top_iff.mpr hI.1.1,
λ J I_lt_J, let ⟨I_le_J, x, hxJ, hxI⟩ := set_like.lt_iff_le_and_exists.mp I_lt_J in
comap_eq_top_iff.1 $ hI.1.2 _ (comap_lt_comap_of_integral_mem_sdiff I_le_J ⟨hxJ, hxI⟩ (hRS x))⟩⟩ | lemma | ideal.is_maximal_of_is_integral_of_is_maximal_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra",
"algebra.is_integral",
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_of_is_integral_of_is_maximal_comap'
(f : R →+* S) (hf : f.is_integral) (I : ideal S) [hI' : I.is_prime]
(hI : is_maximal (I.comap f)) : is_maximal I | @is_maximal_of_is_integral_of_is_maximal_comap R _ S _ f.to_algebra hf I hI' hI | lemma | ideal.is_maximal_of_is_integral_of_is_maximal_comap' | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_of_algebraic_mem [is_domain S] {x : S}
(x_ne_zero : x ≠ 0) (x_mem : x ∈ I) (hx : is_algebraic R x) : I.comap (algebra_map R S) ≠ ⊥ | let ⟨p, p_ne_zero, hp⟩ := hx
in comap_ne_bot_of_root_mem x_ne_zero x_mem p_ne_zero hp | lemma | ideal.comap_ne_bot_of_algebraic_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"is_algebraic",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_bot_of_integral_mem [nontrivial R] [is_domain S] {x : S}
(x_ne_zero : x ≠ 0) (x_mem : x ∈ I) (hx : is_integral R x) : I.comap (algebra_map R S) ≠ ⊥ | comap_ne_bot_of_algebraic_mem x_ne_zero x_mem (hx.is_algebraic R) | lemma | ideal.comap_ne_bot_of_integral_mem | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"is_domain",
"is_integral",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_of_comap_eq_bot [nontrivial R] [is_domain S] (hRS : algebra.is_integral R S)
(hI : I.comap (algebra_map R S) = ⊥) : I = ⊥ | begin
refine eq_bot_iff.2 (λ x hx, _),
by_cases hx0 : x = 0,
{ exact hx0.symm ▸ ideal.zero_mem ⊥ },
{ exact absurd hI (comap_ne_bot_of_integral_mem hx0 hx (hRS x)) }
end | lemma | ideal.eq_bot_of_comap_eq_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra.is_integral",
"algebra_map",
"ideal.zero_mem",
"is_domain",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_comap_of_is_integral_of_is_maximal (hRS : algebra.is_integral R S)
(I : ideal S) [hI : I.is_maximal] : is_maximal (I.comap (algebra_map R S)) | begin
refine quotient.maximal_of_is_field _ _,
haveI : is_prime (I.comap (algebra_map R S)) := comap_is_prime _ _,
exact is_field_of_is_integral_of_is_field (is_integral_quotient_of_is_integral hRS)
algebra_map_quotient_injective (by rwa ← quotient.maximal_ideal_iff_is_field_quotient),
end | lemma | ideal.is_maximal_comap_of_is_integral_of_is_maximal | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra.is_integral",
"algebra_map",
"ideal",
"is_field_of_is_integral_of_is_field",
"is_integral_quotient_of_is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_comap_of_is_integral_of_is_maximal'
{R S : Type*} [comm_ring R] [comm_ring S]
(f : R →+* S) (hf : f.is_integral) (I : ideal S) (hI : I.is_maximal) : is_maximal (I.comap f) | @is_maximal_comap_of_is_integral_of_is_maximal R _ S _ f.to_algebra hf I hI | lemma | ideal.is_maximal_comap_of_is_integral_of_is_maximal' | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.comap_lt_comap {I J : ideal A} [I.is_prime]
(I_lt_J : I < J) :
I.comap (algebra_map R A) < J.comap (algebra_map R A) | let ⟨I_le_J, x, hxJ, hxI⟩ := set_like.lt_iff_le_and_exists.mp I_lt_J in
comap_lt_comap_of_integral_mem_sdiff I_le_J ⟨hxJ, hxI⟩ (is_integral_closure.is_integral R S x) | lemma | ideal.is_integral_closure.comap_lt_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"is_integral_closure.is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.is_maximal_of_is_maximal_comap
(I : ideal A) [I.is_prime]
(hI : is_maximal (I.comap (algebra_map R A))) : is_maximal I | is_maximal_of_is_integral_of_is_maximal_comap (λ x, is_integral_closure.is_integral R S x) I hI | lemma | ideal.is_integral_closure.is_maximal_of_is_maximal_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"is_integral_closure.is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.comap_ne_bot [nontrivial R] {I : ideal A}
(I_ne_bot : I ≠ ⊥) : I.comap (algebra_map R A) ≠ ⊥ | let ⟨x, x_mem, x_ne_zero⟩ := I.ne_bot_iff.mp I_ne_bot in
comap_ne_bot_of_integral_mem x_ne_zero x_mem (is_integral_closure.is_integral R S x) | lemma | ideal.is_integral_closure.comap_ne_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"is_integral_closure.is_integral",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.eq_bot_of_comap_eq_bot [nontrivial R] {I : ideal A} :
I.comap (algebra_map R A) = ⊥ → I = ⊥ | imp_of_not_imp_not _ _ (is_integral_closure.comap_ne_bot S) | lemma | ideal.is_integral_closure.eq_bot_of_comap_eq_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"imp_of_not_imp_not",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.comap_lt_comap {I J : ideal (integral_closure R S)} [I.is_prime]
(I_lt_J : I < J) :
I.comap (algebra_map R (integral_closure R S)) < J.comap (algebra_map R (integral_closure R S)) | is_integral_closure.comap_lt_comap S I_lt_J | lemma | ideal.integral_closure.comap_lt_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.is_maximal_of_is_maximal_comap
(I : ideal (integral_closure R S)) [I.is_prime]
(hI : is_maximal (I.comap (algebra_map R (integral_closure R S)))) : is_maximal I | is_integral_closure.is_maximal_of_is_maximal_comap S I hI | lemma | ideal.integral_closure.is_maximal_of_is_maximal_comap | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.comap_ne_bot [nontrivial R] {I : ideal (integral_closure R S)}
(I_ne_bot : I ≠ ⊥) : I.comap (algebra_map R (integral_closure R S)) ≠ ⊥ | is_integral_closure.comap_ne_bot S I_ne_bot | lemma | ideal.integral_closure.comap_ne_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"integral_closure",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.eq_bot_of_comap_eq_bot [nontrivial R] {I : ideal (integral_closure R S)} :
I.comap (algebra_map R (integral_closure R S)) = ⊥ → I = ⊥ | is_integral_closure.eq_bot_of_comap_eq_bot S | lemma | ideal.integral_closure.eq_bot_of_comap_eq_bot | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra_map",
"ideal",
"integral_closure",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ideal_over_prime_of_is_integral' (H : algebra.is_integral R S)
(P : ideal R) [is_prime P] (hP : (algebra_map R S).ker ≤ P) :
∃ (Q : ideal S), is_prime Q ∧ Q.comap (algebra_map R S) = P | begin
have hP0 : (0 : S) ∉ algebra.algebra_map_submonoid S P.prime_compl,
{ rintro ⟨x, ⟨hx, x0⟩⟩,
exact absurd (hP x0) hx },
let Rₚ := localization P.prime_compl,
let Sₚ := localization (algebra.algebra_map_submonoid S P.prime_compl),
letI : is_domain (localization (algebra.algebra_map_submonoid S P.prime... | lemma | ideal.exists_ideal_over_prime_of_is_integral' | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra.algebra_map_submonoid",
"algebra.is_integral",
"algebra_map",
"ideal",
"is_domain",
"is_integral_localization",
"is_localization.is_domain_localization",
"is_localization.map_comp",
"le_non_zero_divisors_of_no_zero_divisors",
"local_ring.eq_maximal_ideal",
"localization",
"localizatio... | `comap (algebra_map R S)` is a surjection from the prime spec of `R` to prime spec of `S`.
`hP : (algebra_map R S).ker ≤ P` is a slight generalization of the extension being injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ideal_over_prime_of_is_integral (H : algebra.is_integral R S)
(P : ideal R) [is_prime P] (I : ideal S) [is_prime I] (hIP : I.comap (algebra_map R S) ≤ P) :
∃ Q ≥ I, is_prime Q ∧ Q.comap (algebra_map R S) = P | begin
let quot := (R ⧸ I.comap (algebra_map R S)),
obtain ⟨Q' : ideal (S ⧸ I), ⟨Q'_prime, hQ'⟩⟩ :=
@exists_ideal_over_prime_of_is_integral'
quot _ (S ⧸ I) _
ideal.quotient_algebra
_
(is_integral_quotient_of_is_integral H)
(map (quotient.mk (I.comap (algebra_map R S))) P)
(map... | theorem | ideal.exists_ideal_over_prime_of_is_integral | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra.is_integral",
"algebra_map",
"bot_le",
"ideal",
"ideal.quotient_algebra",
"is_integral_quotient_of_is_integral",
"ring_hom.injective_iff_ker_eq_bot",
"ring_hom.ker_eq_comap_bot"
] | More general going-up theorem than `exists_ideal_over_prime_of_is_integral'`.
TODO: Version of going-up theorem with arbitrary length chains (by induction on this)?
Not sure how best to write an ascending chain in Lean | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ideal_over_maximal_of_is_integral [is_domain S] (H : algebra.is_integral R S)
(P : ideal R) [P_max : is_maximal P] (hP : (algebra_map R S).ker ≤ P) :
∃ (Q : ideal S), is_maximal Q ∧ Q.comap (algebra_map R S) = P | begin
obtain ⟨Q, ⟨Q_prime, hQ⟩⟩ := exists_ideal_over_prime_of_is_integral' H P hP,
haveI : Q.is_prime := Q_prime,
exact ⟨Q, is_maximal_of_is_integral_of_is_maximal_comap H _ (hQ.symm ▸ P_max), hQ⟩,
end | lemma | ideal.exists_ideal_over_maximal_of_is_integral | ring_theory.ideal | src/ring_theory/ideal/over.lean | [
"ring_theory.algebraic",
"ring_theory.localization.at_prime",
"ring_theory.localization.integral"
] | [
"algebra.is_integral",
"algebra_map",
"ideal",
"is_domain"
] | `comap (algebra_map R S)` is a surjection from the max spec of `S` to max spec of `R`.
`hP : (algebra_map R S).ker ≤ P` is a slight generalization of the extension being injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod : ideal (R × S) | { carrier := { x | x.fst ∈ I ∧ x.snd ∈ J },
zero_mem' := by simp,
add_mem' :=
begin
rintros ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ⟨ha₁, ha₂⟩ ⟨hb₁, hb₂⟩,
exact ⟨I.add_mem ha₁ hb₁, J.add_mem ha₂ hb₂⟩
end,
smul_mem' :=
begin
rintros ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ⟨hb₁, hb₂⟩,
exact ⟨I.mul_mem_left _ hb₁, J.mul_mem_left _ hb₂⟩,
... | def | ideal.prod | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal"
] | `I × J` as an ideal of `R × S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_prod {r : R} {s : S} : (⟨r, s⟩ : R × S) ∈ prod I J ↔ r ∈ I ∧ s ∈ J | iff.rfl | lemma | ideal.mem_prod | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top_top : prod (⊤ : ideal R) (⊤ : ideal S) = ⊤ | ideal.ext $ by simp | lemma | ideal.prod_top_top | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_prod_eq (I : ideal (R × S)) :
I = ideal.prod (map (ring_hom.fst R S) I) (map (ring_hom.snd R S) I) | begin
apply ideal.ext,
rintro ⟨r, s⟩,
rw [mem_prod, mem_map_iff_of_surjective (ring_hom.fst R S) prod.fst_surjective,
mem_map_iff_of_surjective (ring_hom.snd R S) prod.snd_surjective],
refine ⟨λ h, ⟨⟨_, ⟨h, rfl⟩⟩, ⟨_, ⟨h, rfl⟩⟩⟩, _⟩,
rintro ⟨⟨⟨r, s'⟩, ⟨h₁, rfl⟩⟩, ⟨⟨r', s⟩, ⟨h₂, rfl⟩⟩⟩,
simpa using I.add... | theorem | ideal.ideal_prod_eq | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.ext",
"ideal.prod",
"prod.fst_surjective",
"prod.snd_surjective",
"ring_hom.fst",
"ring_hom.snd"
] | Every ideal of the product ring is of the form `I × J`, where `I` and `J` can be explicitly
given as the image under the projection maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_fst_prod (I : ideal R) (J : ideal S) : map (ring_hom.fst R S) (prod I J) = I | begin
ext,
rw mem_map_iff_of_surjective (ring_hom.fst R S) prod.fst_surjective,
exact ⟨by { rintro ⟨x, ⟨h, rfl⟩⟩, exact h.1 }, λ h, ⟨⟨x, 0⟩, ⟨⟨h, ideal.zero_mem _⟩, rfl⟩⟩⟩
end | lemma | ideal.map_fst_prod | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.zero_mem",
"prod.fst_surjective",
"ring_hom.fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_snd_prod (I : ideal R) (J : ideal S) : map (ring_hom.snd R S) (prod I J) = J | begin
ext,
rw mem_map_iff_of_surjective (ring_hom.snd R S) prod.snd_surjective,
exact ⟨by { rintro ⟨x, ⟨h, rfl⟩⟩, exact h.2 }, λ h, ⟨⟨0, x⟩, ⟨⟨ideal.zero_mem _, h⟩, rfl⟩⟩⟩
end | lemma | ideal.map_snd_prod | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"prod.snd_surjective",
"ring_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod_comm_prod :
map ((ring_equiv.prod_comm : R × S ≃+* S × R) : R × S →+* S × R) (prod I J) = prod J I | begin
refine trans (ideal_prod_eq _) _,
simp [map_map],
end | lemma | ideal.map_prod_comm_prod | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ring_equiv.prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_prod_equiv : ideal (R × S) ≃ ideal R × ideal S | { to_fun := λ I, ⟨map (ring_hom.fst R S) I, map (ring_hom.snd R S) I⟩,
inv_fun := λ I, prod I.1 I.2,
left_inv := λ I, (ideal_prod_eq I).symm,
right_inv := λ ⟨I, J⟩, by simp } | def | ideal.ideal_prod_equiv | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"inv_fun",
"ring_hom.fst",
"ring_hom.snd"
] | Ideals of `R × S` are in one-to-one correspondence with pairs of ideals of `R` and ideals of
`S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_prod_equiv_symm_apply (I : ideal R) (J : ideal S) :
ideal_prod_equiv.symm ⟨I, J⟩ = prod I J | rfl | lemma | ideal.ideal_prod_equiv_symm_apply | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.ext_iff {I I' : ideal R} {J J' : ideal S} : prod I J = prod I' J' ↔ I = I' ∧ J = J' | by simp only [←ideal_prod_equiv_symm_apply, ideal_prod_equiv.symm.injective.eq_iff, prod.mk.inj_iff] | lemma | ideal.prod.ext_iff | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"prod.ext_iff",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_of_is_prime_prod_top {I : ideal R} (h : (ideal.prod I (⊤ : ideal S)).is_prime) :
I.is_prime | begin
split,
{ unfreezingI { contrapose! h },
simp [is_prime_iff, h] },
{ intros x y hxy,
have : (⟨x, 1⟩ : R × S) * ⟨y, 1⟩ ∈ prod I ⊤,
{ rw [prod.mk_mul_mk, mul_one, mem_prod],
exact ⟨hxy, trivial⟩ },
simpa using h.mem_or_mem this }
end | lemma | ideal.is_prime_of_is_prime_prod_top | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.prod",
"mul_one",
"prod.mk_mul_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_of_is_prime_prod_top' {I : ideal S} (h : (ideal.prod (⊤ : ideal R) I).is_prime) :
I.is_prime | begin
apply @is_prime_of_is_prime_prod_top _ R,
rw ←map_prod_comm_prod,
exact map_is_prime_of_equiv _
end | lemma | ideal.is_prime_of_is_prime_prod_top' | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_ideal_prod_top {I : ideal R} [h : I.is_prime] : (prod I (⊤ : ideal S)).is_prime | begin
split,
{ unfreezingI { rcases h with ⟨h, -⟩, contrapose! h },
rw [←prod_top_top, prod.ext_iff] at h,
exact h.1 },
rintros ⟨r₁, s₁⟩ ⟨r₂, s₂⟩ ⟨h₁, h₂⟩,
cases h.mem_or_mem h₁ with h h,
{ exact or.inl ⟨h, trivial⟩ },
{ exact or.inr ⟨h, trivial⟩ }
end | lemma | ideal.is_prime_ideal_prod_top | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"prod.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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