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
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|---|---|---|---|---|---|---|---|---|---|---|
is_prime_ideal_prod_top' {I : ideal S} [h : I.is_prime] : (prod (⊤ : ideal R) I).is_prime | begin
rw ←map_prod_comm_prod,
apply map_is_prime_of_equiv _,
exact is_prime_ideal_prod_top,
end | lemma | ideal.is_prime_ideal_prod_top' | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_prod_prime_aux {I : ideal R} {J : ideal S} : (ideal.prod I J).is_prime →
I = ⊤ ∨ J = ⊤ | begin
contrapose!,
simp only [ne_top_iff_one, is_prime_iff, not_and, not_forall, not_or_distrib],
exact λ ⟨hI, hJ⟩ hIJ, ⟨⟨0, 1⟩, ⟨1, 0⟩, by simp, by simp [hJ], by simp [hI]⟩
end | lemma | ideal.ideal_prod_prime_aux | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.prod",
"not_and",
"not_forall",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_prod_prime (I : ideal (R × S)) : I.is_prime ↔
((∃ p : ideal R, p.is_prime ∧ I = ideal.prod p ⊤) ∨
(∃ p : ideal S, p.is_prime ∧ I = ideal.prod ⊤ p)) | begin
split,
{ rw ideal_prod_eq I,
introsI hI,
rcases ideal_prod_prime_aux hI with (h|h),
{ right,
rw h at hI ⊢,
exact ⟨_, ⟨is_prime_of_is_prime_prod_top' hI, rfl⟩⟩ },
{ left,
rw h at hI ⊢,
exact ⟨_, ⟨is_prime_of_is_prime_prod_top hI, rfl⟩⟩ } },
{ rintro (⟨p, ⟨h, rfl⟩⟩|⟨p, ... | theorem | ideal.ideal_prod_prime | ring_theory.ideal | src/ring_theory/ideal/prod.lean | [
"ring_theory.ideal.operations"
] | [
"ideal",
"ideal.prod"
] | Classification of prime ideals in product rings: the prime ideals of `R × S` are precisely the
ideals of the form `p × S` or `R × p`, where `p` is a prime ideal of `R` or `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_one (I : ideal R) : has_one (R ⧸ I) | ⟨submodule.quotient.mk 1⟩ | instance | ideal.quotient.has_one | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_con (I : ideal R) : ring_con R | { mul' := λ a₁ b₁ a₂ b₂ h₁ h₂, begin
rw submodule.quotient_rel_r_def at h₁ h₂ ⊢,
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂),
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁,
{ rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁] },
rw ← this at F,
c... | def | ideal.quotient.ring_con | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"mul_comm",
"ring_con",
"submodule.quotient_rel_r_def"
] | On `ideal`s, `submodule.quotient_rel` is a ring congruence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring (I : ideal R) : comm_ring (R ⧸ I) | { ..submodule.quotient.add_comm_group I, -- to help with unification
..(quotient.ring_con I)^.quotient.comm_ring } | instance | ideal.quotient.comm_ring | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"comm_ring",
"ideal",
"submodule.quotient.add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_right {α} [has_smul α R] [is_scalar_tower α R R] :
is_scalar_tower α (R ⧸ I) (R ⧸ I) | (quotient.ring_con I)^.is_scalar_tower_right | instance | ideal.quotient.is_scalar_tower_right | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"has_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class {α} [has_smul α R] [is_scalar_tower α R R] [smul_comm_class α R R] :
smul_comm_class α (R ⧸ I) (R ⧸ I) | (quotient.ring_con I)^.smul_comm_class | instance | ideal.quotient.smul_comm_class | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"has_smul",
"is_scalar_tower",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class' {α} [has_smul α R] [is_scalar_tower α R R] [smul_comm_class R α R] :
smul_comm_class (R ⧸ I) α (R ⧸ I) | (quotient.ring_con I)^.smul_comm_class' | instance | ideal.quotient.smul_comm_class' | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"has_smul",
"is_scalar_tower",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (I : ideal R) : R →+* (R ⧸ I) | ⟨λ a, submodule.quotient.mk a, rfl, λ _ _, rfl, rfl, λ _ _, rfl⟩ | def | ideal.quotient.mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"submodule.quotient.mk"
] | The ring homomorphism from a ring `R` to a quotient ring `R/I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_ext [non_assoc_semiring S] ⦃f g : R ⧸ I →+* S⦄
(h : f.comp (mk I) = g.comp (mk I)) : f = g | ring_hom.ext $ λ x, quotient.induction_on' x $ (ring_hom.congr_fun h : _) | lemma | ideal.quotient.ring_hom_ext | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"non_assoc_semiring",
"quotient.induction_on'",
"ring_hom.congr_fun",
"ring_hom.ext",
"ring_hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited : inhabited (R ⧸ I) | ⟨mk I 37⟩ | instance | ideal.quotient.inhabited | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq : mk I x = mk I y ↔ x - y ∈ I | submodule.quotient.eq I | theorem | ideal.quotient.eq | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"submodule.quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk (x : R) : (submodule.quotient.mk x : R ⧸ I) = mk I x | rfl | theorem | ideal.quotient.mk_eq_mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"submodule.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff_mem {I : ideal R} : mk I a = 0 ↔ a ∈ I | submodule.quotient.mk_eq_zero _ | lemma | ideal.quotient.eq_zero_iff_mem | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"submodule.quotient.mk_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_one_iff {I : ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ | eq_comm.trans $ eq_zero_iff_mem.trans (eq_top_iff_one _).symm | theorem | ideal.quotient.zero_eq_one_iff | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_one_iff {I : ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ | not_congr zero_eq_one_iff | theorem | ideal.quotient.zero_ne_one_iff | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial {I : ideal R} (hI : I ≠ ⊤) : nontrivial (R ⧸ I) | ⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩ | theorem | ideal.quotient.nontrivial | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_iff {I : ideal R} : subsingleton (R ⧸ I) ↔ I = ⊤ | by rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm,
← I^.quotient.mk^.map_one, quotient.eq_zero_iff_mem] | lemma | ideal.quotient.subsingleton_iff | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"map_one",
"subsingleton_iff",
"subsingleton_iff_zero_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_surjective : function.surjective (mk I) | λ y, quotient.induction_on' y (λ x, exists.intro x rfl) | lemma | ideal.quotient.mk_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"quotient.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ring_saturate (I : ideal R) (s : set R) :
mk I ⁻¹' (mk I '' s) = (⋃ x : I, (λ y, x.1 + y) '' s) | begin
ext x,
simp only [mem_preimage, mem_image, mem_Union, ideal.quotient.eq],
exact ⟨λ ⟨a, a_in, h⟩, ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩,
λ ⟨⟨i, hi⟩, a, ha, eq⟩,
⟨a, ha, by rw [← eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
end | lemma | ideal.quotient.quotient_ring_saturate | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.eq"
] | If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors (I : ideal R) [hI : I.is_prime] : no_zero_divisors (R ⧸ I) | { eq_zero_or_eq_zero_of_mul_eq_zero := λ a b,
quotient.induction_on₂' a b $ λ a b hab,
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim
(or.inl ∘ eq_zero_iff_mem.2)
(or.inr ∘ eq_zero_iff_mem.2) } | instance | ideal.quotient.no_zero_divisors | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"no_zero_divisors",
"quotient.induction_on₂'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain (I : ideal R) [hI : I.is_prime] : is_domain (R ⧸ I) | let _ := quotient.nontrivial hI.1 in by exactI no_zero_divisors.to_is_domain _ | instance | ideal.quotient.is_domain | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"is_domain",
"no_zero_divisors.to_is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain_iff_prime (I : ideal R) : is_domain (R ⧸ I) ↔ I.is_prime | begin
refine ⟨λ H, ⟨zero_ne_one_iff.1 _, λ x y h, _⟩, λ h, by { resetI, apply_instance }⟩,
{ haveI : nontrivial (R ⧸ I) := ⟨H.3⟩,
exact zero_ne_one },
{ simp only [←eq_zero_iff_mem, (mk I).map_mul] at ⊢ h,
haveI := @is_domain.to_no_zero_divisors (R ⧸ I) _ H,
exact eq_zero_or_eq_zero_of_mul_eq_zero h }... | lemma | ideal.quotient.is_domain_iff_prime | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"is_domain",
"is_domain.to_no_zero_divisors",
"map_mul",
"nontrivial",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_inv {I : ideal R} [hI : I.is_maximal] :
∀ {a : (R ⧸ I)}, a ≠ 0 → ∃ b : (R ⧸ I), a * b = 1 | begin
rintro ⟨a⟩ h,
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩,
rw [mul_comm] at abc,
refine ⟨mk _ b, quot.sound _⟩, --quot.sound hb
rw ← eq_sub_iff_add_eq' at abc,
rw [abc, ← neg_mem_iff, neg_sub] at hc,
rw submodule.quotient_rel_r_def,
convert hc,
end | lemma | ideal.quotient.exists_inv | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"mul_comm",
"submodule.quotient_rel_r_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero (I : ideal R) [hI : I.is_maximal] :
group_with_zero (R ⧸ I) | { inv := λ a, if ha : a = 0 then 0 else classical.some (exists_inv ha),
mul_inv_cancel := λ a (ha : a ≠ 0), show a * dite _ _ _ = _,
by rw dif_neg ha;
exact classical.some_spec (exists_inv ha),
inv_zero := dif_pos rfl,
..(by apply_instance : monoid_with_zero (R ⧸ I)),
..quotient.is_domain I } | def | ideal.quotient.group_with_zero | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"group_with_zero",
"ideal",
"inv_zero",
"monoid_with_zero",
"mul_inv_cancel"
] | The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field (I : ideal R) [hI : I.is_maximal] : field (R ⧸ I) | { ..quotient.comm_ring I,
..quotient.group_with_zero I } | def | ideal.quotient.field | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"field",
"ideal"
] | The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_of_is_field (I : ideal R)
(hqf : is_field (R ⧸ I)) : I.is_maximal | begin
apply ideal.is_maximal_iff.2,
split,
{ intro h,
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩,
exact hxy (ideal.quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h)) },
{ intros J x hIJ hxnI hxJ,
rcases hqf.mul_inv_cancel (mt ideal.quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩,
rw [← ze... | theorem | ideal.quotient.maximal_of_is_field | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"is_field",
"mul_one"
] | If the quotient by an ideal is a field, then the ideal is maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_ideal_iff_is_field_quotient (I : ideal R) : I.is_maximal ↔ is_field (R ⧸ I) | ⟨λ h, by { letI := @quotient.field _ _ I h, exact field.to_is_field _ }, maximal_of_is_field _⟩ | theorem | ideal.quotient.maximal_ideal_iff_is_field_quotient | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"field.to_is_field",
"ideal",
"is_field"
] | The quotient of a ring by an ideal is a field iff the ideal is maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift (I : ideal R) (f : R →+* S) (H : ∀ (a : R), a ∈ I → f a = 0) :
R ⧸ I →+* S | { map_one' := f.map_one,
map_zero' := f.map_zero,
map_add' := λ a₁ a₂, quotient.induction_on₂' a₁ a₂ f.map_add,
map_mul' := λ a₁ a₂, quotient.induction_on₂' a₁ a₂ f.map_mul,
.. quotient_add_group.lift I.to_add_subgroup f.to_add_monoid_hom H } | def | ideal.quotient.lift | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"lift",
"quotient.induction_on₂'"
] | Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk (I : ideal R) (f : R →+* S) (H : ∀ (a : R), a ∈ I → f a = 0) :
lift I f H (mk I a) = f a | rfl | lemma | ideal.quotient.lift_mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"lift",
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_surjective_of_surjective (I : ideal R) {f : R →+* S} (H : ∀ (a : R), a ∈ I → f a = 0)
(hf : function.surjective f) : function.surjective (ideal.quotient.lift I f H) | begin
intro y,
obtain ⟨x, rfl⟩ := hf y,
use ideal.quotient.mk I x,
simp only [ideal.quotient.lift_mk],
end | lemma | ideal.quotient.lift_surjective_of_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor (S T : ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T | ideal.quotient.lift S (T^.quotient.mk) (λ x hx, eq_zero_iff_mem.2 (H hx)) | def | ideal.quotient.factor | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.lift"
] | The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `ideal.quotient` version of `quot.factor` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factor_mk (S T : ideal R) (H : S ≤ T) (x : R) :
factor S T H (mk S x) = mk T x | rfl | lemma | ideal.quotient.factor_mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_comp_mk (S T : ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T | by { ext x, rw [ring_hom.comp_apply, factor_mk] } | lemma | ideal.quotient.factor_comp_mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_equiv_of_eq {R : Type*} [comm_ring R] {I J : ideal R} (h : I = J) :
(R ⧸ I) ≃+* R ⧸ J | { map_mul' := by { rintro ⟨x⟩ ⟨y⟩, refl },
.. submodule.quot_equiv_of_eq I J h } | def | ideal.quot_equiv_of_eq | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"comm_ring",
"ideal",
"submodule.quot_equiv_of_eq"
] | Quotienting by equal ideals gives equivalent rings.
See also `submodule.quot_equiv_of_eq` and `ideal.quotient_equiv_alg_of_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_equiv_of_eq_mk {R : Type*} [comm_ring R] {I J : ideal R} (h : I = J) (x : R) :
quot_equiv_of_eq h (ideal.quotient.mk I x) = ideal.quotient.mk J x | rfl | lemma | ideal.quot_equiv_of_eq_mk | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"comm_ring",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_equiv_of_eq_symm {R : Type*} [comm_ring R] {I J : ideal R} (h : I = J) :
(ideal.quot_equiv_of_eq h).symm = ideal.quot_equiv_of_eq h.symm | by ext; refl | lemma | ideal.quot_equiv_of_eq_symm | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"comm_ring",
"ideal",
"ideal.quot_equiv_of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_pi : module (R ⧸ I) ((ι → R) ⧸ I.pi ι) | { smul := λ c m, quotient.lift_on₂' c m (λ r m, submodule.quotient.mk $ r • m) begin
intros c₁ m₁ c₂ m₂ hc hm,
apply ideal.quotient.eq.2,
rw submodule.quotient_rel_r_def at hc hm,
intro i,
exact I.mul_sub_mul_mem hc (hm i),
end,
one_smul := begin
rintro ⟨a⟩,
convert_to ideal.quotient.mk ... | instance | ideal.module_pi | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"add_smul",
"ideal.quotient.mk",
"module",
"mul_assoc",
"mul_zero",
"one_mul",
"one_smul",
"quotient.lift_on₂'",
"smul_add",
"smul_zero",
"submodule.quotient.mk",
"submodule.quotient_rel_r_def",
"zero_mul",
"zero_smul"
] | `R^n/I^n` is a `R/I`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_quot_equiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[(R ⧸ I)] (ι → (R ⧸ I)) | { to_fun := λ x, quotient.lift_on' x (λ f i, ideal.quotient.mk I (f i)) $
λ a b hab, funext (λ i, (submodule.quotient.eq' _).2
(quotient_add_group.left_rel_apply.mp hab i)),
map_add' := by { rintros ⟨_⟩ ⟨_⟩, refl },
map_smul' := by { rintros ⟨_⟩ ⟨_⟩, refl },
inv_fun := λ x, ideal.quotient.mk (I.pi ι) $ ... | def | ideal.pi_quot_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal.quotient.mk",
"inv_fun",
"quotient.lift_on'",
"quotient.out'",
"quotient.out_eq'",
"submodule.quotient.eq'"
] | `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pi {ι : Type*} [finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
(f : (ι → R) →ₗ[R] (ι' → R)) (i : ι') : f x i ∈ I | begin
classical,
casesI nonempty_fintype ι,
rw pi_eq_sum_univ x,
simp only [finset.sum_apply, smul_eq_mul, linear_map.map_sum, pi.smul_apply, linear_map.map_smul],
exact I.sum_mem (λ j hj, I.mul_mem_right _ (hi j))
end | lemma | ideal.map_pi | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"finite",
"linear_map.map_smul",
"linear_map.map_sum",
"nonempty_fintype",
"pi.smul_apply",
"pi_eq_sum_univ",
"smul_eq_mul"
] | If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
contained in `I^m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_sub_one_mem_and_mem (s : finset ι) {f : ι → ideal R}
(hf : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ⊔ f j = ⊤) (i : ι) (his : i ∈ s) :
∃ r : R, r - 1 ∈ f i ∧ ∀ j ∈ s, j ≠ i → r ∈ f j | begin
have : ∀ j ∈ s, j ≠ i → ∃ r : R, ∃ H : r - 1 ∈ f i, r ∈ f j,
{ intros j hjs hji, specialize hf i his j hjs hji.symm,
rw [eq_top_iff_one, submodule.mem_sup] at hf,
rcases hf with ⟨r, hri, s, hsj, hrs⟩, refine ⟨1 - r, _, _⟩,
{ rw [sub_right_comm, sub_self, zero_sub], exact (f i).neg_mem hri },
{... | theorem | ideal.exists_sub_one_mem_and_mem | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"finset",
"finset.mem_erase_of_ne_of_mem",
"finset.prod_eq_one",
"finset.prod_eq_zero",
"ideal",
"quotient.eq",
"ring_hom.map_one",
"ring_hom.map_prod",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_sub_mem [finite ι] {f : ι → ideal R} (hf : ∀ i j, i ≠ j → f i ⊔ f j = ⊤)
(g : ι → R) :
∃ r : R, ∀ i, r - g i ∈ f i | begin
casesI nonempty_fintype ι,
have : ∃ φ : ι → R, (∀ i, φ i - 1 ∈ f i) ∧ (∀ i j, i ≠ j → φ i ∈ f j),
{ have := exists_sub_one_mem_and_mem (finset.univ : finset ι) (λ i _ j _ hij, hf i j hij),
choose φ hφ,
existsi λ i, φ i (finset.mem_univ i),
exact ⟨λ i, (hφ i _).1, λ i j hij, (hφ i _).2 j (finset.... | theorem | ideal.exists_sub_mem | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"finite",
"finset",
"finset.mem_univ",
"finset.univ",
"ideal",
"mul_one",
"nonempty_fintype",
"quotient.eq",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_inf_to_pi_quotient (f : ι → ideal R) :
R ⧸ (⨅ i, f i) →+* Π i, R ⧸ f i | quotient.lift (⨅ i, f i)
(pi.ring_hom (λ i : ι, (quotient.mk (f i) : _))) $
λ r hr, begin
rw submodule.mem_infi at hr,
ext i,
exact quotient.eq_zero_iff_mem.2 (hr i)
end | def | ideal.quotient_inf_to_pi_quotient | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"pi.ring_hom",
"submodule.mem_infi"
] | The homomorphism from `R/(⋂ i, f i)` to `∏ i, (R / f i)` featured in the Chinese
Remainder Theorem. It is bijective if the ideals `f i` are comaximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_inf_to_pi_quotient_bijective [finite ι] {f : ι → ideal R}
(hf : ∀ i j, i ≠ j → f i ⊔ f j = ⊤) :
function.bijective (quotient_inf_to_pi_quotient f) | ⟨λ x y, quotient.induction_on₂' x y $ λ r s hrs, quotient.eq.2 $
(submodule.mem_infi _).2 $ λ i, quotient.eq.1 $
show quotient_inf_to_pi_quotient f (quotient.mk' r) i = _, by rw hrs; refl,
λ g, let ⟨r, hr⟩ := exists_sub_mem hf (λ i, quotient.out' (g i)) in
⟨quotient.mk _ r, funext $ λ i, quotient.out_eq' (g i) ▸ qu... | theorem | ideal.quotient_inf_to_pi_quotient_bijective | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"finite",
"ideal",
"quotient.induction_on₂'",
"quotient.mk'",
"quotient.out'",
"quotient.out_eq'",
"submodule.mem_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_inf_ring_equiv_pi_quotient [finite ι] (f : ι → ideal R)
(hf : ∀ i j, i ≠ j → f i ⊔ f j = ⊤) :
R ⧸ (⨅ i, f i) ≃+* Π i, R ⧸ f i | { .. equiv.of_bijective _ (quotient_inf_to_pi_quotient_bijective hf),
.. quotient_inf_to_pi_quotient f } | def | ideal.quotient_inf_ring_equiv_pi_quotient | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"equiv.of_bijective",
"finite",
"ideal"
] | Chinese Remainder Theorem. Eisenbud Ex.2.6. Similar to Atiyah-Macdonald 1.10 and Stacks 00DT | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_inf_equiv_quotient_prod (I J : ideal R)
(coprime : I ⊔ J = ⊤) :
(R ⧸ (I ⊓ J)) ≃+* (R ⧸ I) × R ⧸ J | let f : fin 2 → ideal R := ![I, J] in
have hf : ∀ (i j : fin 2), i ≠ j → f i ⊔ f j = ⊤,
by { intros i j h,
fin_cases i; fin_cases j; try { contradiction }; simpa [f, sup_comm] using coprime },
(ideal.quot_equiv_of_eq (by simp [infi, inf_comm])).trans $
(ideal.quotient_inf_ring_equiv_pi_quotient f hf).trans $
ring_equ... | def | ideal.quotient_inf_equiv_quotient_prod | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quot_equiv_of_eq",
"ideal.quotient_inf_ring_equiv_pi_quotient",
"inf_comm",
"infi",
"ring_equiv.pi_fin_two",
"sup_comm"
] | **Chinese remainder theorem**, specialized to two ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_inf_equiv_quotient_prod_fst (I J : ideal R) (coprime : I ⊔ J = ⊤)
(x : R ⧸ (I ⊓ J)) : (quotient_inf_equiv_quotient_prod I J coprime x).fst =
ideal.quotient.factor (I ⊓ J) I inf_le_left x | quot.induction_on x (λ x, rfl) | lemma | ideal.quotient_inf_equiv_quotient_prod_fst | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.factor",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_inf_equiv_quotient_prod_snd (I J : ideal R) (coprime : I ⊔ J = ⊤)
(x : R ⧸ (I ⊓ J)) : (quotient_inf_equiv_quotient_prod I J coprime x).snd =
ideal.quotient.factor (I ⊓ J) J inf_le_right x | quot.induction_on x (λ x, rfl) | lemma | ideal.quotient_inf_equiv_quotient_prod_snd | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.factor",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_quotient_inf_equiv_quotient_prod (I J : ideal R) (coprime : I ⊔ J = ⊤) :
(ring_hom.fst _ _).comp
(quotient_inf_equiv_quotient_prod I J coprime : R ⧸ I ⊓ J →+* (R ⧸ I) × R ⧸ J) =
ideal.quotient.factor (I ⊓ J) I inf_le_left | by ext; refl | lemma | ideal.fst_comp_quotient_inf_equiv_quotient_prod | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.factor",
"inf_le_left",
"ring_hom.fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_quotient_inf_equiv_quotient_prod (I J : ideal R) (coprime : I ⊔ J = ⊤) :
(ring_hom.snd _ _).comp
(quotient_inf_equiv_quotient_prod I J coprime : R ⧸ I ⊓ J →+* (R ⧸ I) × R ⧸ J) =
ideal.quotient.factor (I ⊓ J) J inf_le_right | by ext; refl | lemma | ideal.snd_comp_quotient_inf_equiv_quotient_prod | ring_theory.ideal | src/ring_theory/ideal/quotient.lean | [
"algebra.ring.fin",
"algebra.ring.prod",
"linear_algebra.quotient",
"ring_theory.congruence",
"ring_theory.ideal.basic",
"tactic.fin_cases"
] | [
"ideal",
"ideal.quotient.factor",
"inf_le_right",
"ring_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift (f : R →+* S) : R ⧸ f.ker →+* S | ideal.quotient.lift _ f $ λ r, f.mem_ker.mp | def | ring_hom.ker_lift | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.lift"
] | The induced map from the quotient by the kernel to the codomain.
This is an isomorphism if `f` has a right inverse (`quotient_ker_equiv_of_right_inverse`) /
is surjective (`quotient_ker_equiv_of_surjective`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_lift_mk (f : R →+* S) (r : R) : ker_lift f (ideal.quotient.mk f.ker r) = f r | ideal.quotient.lift_mk _ _ _ | lemma | ring_hom.ker_lift_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.lift_mk",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift_injective (f : R →+* S) : function.injective (ker_lift f) | assume a b, quotient.induction_on₂' a b $
assume a b (h : f a = f b), ideal.quotient.eq.2 $
show a - b ∈ ker f, by rw [mem_ker, map_sub, h, sub_self] | lemma | ring_hom.ker_lift_injective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"quotient.induction_on₂'"
] | The induced map from the quotient by the kernel is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_injective_of_ker_le_ideal (I : ideal R) {f : R →+* S}
(H : ∀ (a : R), a ∈ I → f a = 0) (hI : f.ker ≤ I) :
function.injective (ideal.quotient.lift I f H) | begin
rw [ring_hom.injective_iff_ker_eq_bot, ring_hom.ker_eq_bot_iff_eq_zero],
intros u hu,
obtain ⟨v, rfl⟩ := ideal.quotient.mk_surjective u,
rw ideal.quotient.lift_mk at hu,
rw [ideal.quotient.eq_zero_iff_mem],
exact hI ((ring_hom.mem_ker f).mpr hu),
end | lemma | ring_hom.lift_injective_of_ker_le_ideal | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"ideal.quotient.mk_surjective",
"ring_hom.injective_iff_ker_eq_bot",
"ring_hom.ker_eq_bot_iff_eq_zero",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ker_equiv_of_right_inverse
{g : S → R} (hf : function.right_inverse g f) :
R ⧸ f.ker ≃+* S | { to_fun := ker_lift f,
inv_fun := (ideal.quotient.mk f.ker) ∘ g,
left_inv := begin
rintro ⟨x⟩,
apply ker_lift_injective,
simp [hf (f x)],
end,
right_inv := hf,
..ker_lift f} | def | ring_hom.quotient_ker_equiv_of_right_inverse | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"inv_fun"
] | The **first isomorphism theorem** for commutative rings, computable version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_ker_equiv_of_right_inverse.apply {g : S → R} (hf : function.right_inverse g f)
(x : R ⧸ f.ker) : quotient_ker_equiv_of_right_inverse hf x = ker_lift f x | rfl | lemma | ring_hom.quotient_ker_equiv_of_right_inverse.apply | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ker_equiv_of_right_inverse.symm.apply {g : S → R} (hf : function.right_inverse g f)
(x : S) : (quotient_ker_equiv_of_right_inverse hf).symm x = ideal.quotient.mk f.ker (g x) | rfl | lemma | ring_hom.quotient_ker_equiv_of_right_inverse.symm.apply | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ker_equiv_of_surjective (hf : function.surjective f) :
R ⧸ f.ker ≃+* S | quotient_ker_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse) | def | ring_hom.quotient_ker_equiv_of_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | The **first isomorphism theorem** for commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_quotient_self (I : ideal R) :
map (quotient.mk I) I = ⊥ | eq_bot_iff.2 $ ideal.map_le_iff_le_comap.2 $ λ x hx,
(submodule.mem_bot (R ⧸ I)).2 $ ideal.quotient.eq_zero_iff_mem.2 hx | lemma | ideal.map_quotient_self | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ker {I : ideal R} : (quotient.mk I).ker = I | by ext; rw [ring_hom.ker, mem_comap, submodule.mem_bot, quotient.eq_zero_iff_mem] | lemma | ideal.mk_ker | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ring_hom.ker",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk_eq_bot_of_le {I J : ideal R} (h : I ≤ J) : I.map (J^.quotient.mk) = ⊥ | by { rw [map_eq_bot_iff_le_ker, mk_ker], exact h } | lemma | ideal.map_mk_eq_bot_of_le | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_quotient_lift {S : Type v} [comm_ring S] {I : ideal R} (f : R →+* S) (H : I ≤ f.ker) :
(ideal.quotient.lift I f H).ker = (f.ker).map I^.quotient.mk | begin
ext x,
split,
{ intro hx,
obtain ⟨y, hy⟩ := quotient.mk_surjective x,
rw [ring_hom.mem_ker, ← hy, ideal.quotient.lift_mk, ← ring_hom.mem_ker] at hx,
rw [← hy, mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective],
exact ⟨y, hx, rfl⟩ },
{ intro hx,
rw mem_map_iff_of_surjectiv... | lemma | ideal.ker_quotient_lift | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"comm_ring",
"ideal",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_quotient_is_maximal_iff (I : ideal R) :
(⊥ : ideal (R ⧸ I)).is_maximal ↔ I.is_maximal | ⟨λ hI, (@mk_ker _ _ I) ▸
@comap_is_maximal_of_surjective _ _ _ _ _ _ (quotient.mk I) quotient.mk_surjective ⊥ hI,
λ hI, by { resetI, letI := quotient.field I, exact bot_is_maximal }⟩ | lemma | ideal.bot_quotient_is_maximal_iff | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_quotient_iff_mem_sup {I J : ideal R} {x : R} :
quotient.mk I x ∈ J.map (quotient.mk I) ↔ x ∈ J ⊔ I | by rw [← mem_comap, comap_map_of_surjective (quotient.mk I) quotient.mk_surjective,
← ring_hom.ker_eq_comap_bot, mk_ker] | lemma | ideal.mem_quotient_iff_mem_sup | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ring_hom.ker_eq_comap_bot"
] | See also `ideal.mem_quotient_iff_mem` in case `I ≤ J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_quotient_iff_mem {I J : ideal R} (hIJ : I ≤ J) {x : R} :
quotient.mk I x ∈ J.map (quotient.mk I) ↔ x ∈ J | by rw [mem_quotient_iff_mem_sup, sup_eq_left.mpr hIJ] | lemma | ideal.mem_quotient_iff_mem | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | See also `ideal.mem_quotient_iff_mem_sup` if the assumption `I ≤ J` is not available. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_map_mk {I J : ideal R} (h : I ≤ J) :
ideal.comap (ideal.quotient.mk I) (ideal.map (ideal.quotient.mk I) J) = J | by { ext, rw [← ideal.mem_quotient_iff_mem h, ideal.mem_comap], } | lemma | ideal.comap_map_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.comap",
"ideal.map",
"ideal.mem_comap",
"ideal.mem_quotient_iff_mem",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.algebra {I : ideal A} : algebra R₁ (A ⧸ I) | { to_fun := λ x, ideal.quotient.mk I (algebra_map R₁ A x),
smul := (•),
smul_def' := λ r x, quotient.induction_on' x $ λ x,
((quotient.mk I).congr_arg $ algebra.smul_def _ _).trans (ring_hom.map_mul _ _ _),
commutes' := λ _ _, mul_comm _ _,
.. ring_hom.comp (ideal.quotient.mk I) (algebra_map R₁ A) } | instance | ideal.quotient.algebra | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"ideal",
"ideal.quotient.mk",
"mul_comm",
"quotient.induction_on'",
"ring_hom.comp",
"ring_hom.map_mul"
] | The `R₁`-algebra structure on `A/I` for an `R₁`-algebra `A` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.is_scalar_tower [has_smul R₁ R₂] [is_scalar_tower R₁ R₂ A] (I : ideal A) :
is_scalar_tower R₁ R₂ (A ⧸ I) | by apply_instance | instance | ideal.quotient.is_scalar_tower | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"has_smul",
"ideal",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mkₐ (I : ideal A) : A →ₐ[R₁] A ⧸ I | ⟨λ a, submodule.quotient.mk a, rfl, λ _ _, rfl, rfl, λ _ _, rfl, λ _, rfl⟩ | def | ideal.quotient.mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"submodule.quotient.mk"
] | The canonical morphism `A →ₐ[R₁] A ⧸ I` as morphism of `R₁`-algebras, for `I` an ideal of
`A`, where `A` is an `R₁`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.alg_hom_ext {I : ideal A} {S} [semiring S] [algebra R₁ S] ⦃f g : A ⧸ I →ₐ[R₁] S⦄
(h : f.comp (quotient.mkₐ R₁ I) = g.comp (quotient.mkₐ R₁ I)) : f = g | alg_hom.ext $ λ x, quotient.induction_on' x $ alg_hom.congr_fun h | lemma | ideal.quotient.alg_hom_ext | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.congr_fun",
"alg_hom.ext",
"algebra",
"ideal",
"quotient.induction_on'",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.alg_map_eq (I : ideal A) :
algebra_map R₁ (A ⧸ I) = (algebra_map A (A ⧸ I)).comp (algebra_map R₁ A) | rfl | lemma | ideal.quotient.alg_map_eq | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mkₐ_to_ring_hom (I : ideal A) :
(quotient.mkₐ R₁ I).to_ring_hom = ideal.quotient.mk I | rfl | lemma | ideal.quotient.mkₐ_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mkₐ_eq_mk (I : ideal A) :
⇑(quotient.mkₐ R₁ I) = ideal.quotient.mk I | rfl | lemma | ideal.quotient.mkₐ_eq_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.algebra_map_eq (I : ideal R) :
algebra_map R (R ⧸ I) = I^.quotient.mk | rfl | lemma | ideal.quotient.algebra_map_eq | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mk_comp_algebra_map (I : ideal A) :
(quotient.mk I).comp (algebra_map R₁ A) = algebra_map R₁ (A ⧸ I) | rfl | lemma | ideal.quotient.mk_comp_algebra_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mk_algebra_map (I : ideal A) (x : R₁) :
quotient.mk I (algebra_map R₁ A x) = algebra_map R₁ (A ⧸ I) x | rfl | lemma | ideal.quotient.mk_algebra_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.mkₐ_surjective (I : ideal A) : function.surjective (quotient.mkₐ R₁ I) | surjective_quot_mk _ | lemma | ideal.quotient.mkₐ_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"surjective_quot_mk"
] | The canonical morphism `A →ₐ[R₁] I.quotient` is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.mkₐ_ker (I : ideal A) : (quotient.mkₐ R₁ I : A →+* A ⧸ I).ker = I | ideal.mk_ker | lemma | ideal.quotient.mkₐ_ker | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.mk_ker"
] | The kernel of `A →ₐ[R₁] I.quotient` is `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.liftₐ (I : ideal A) (f : A →ₐ[R₁] B) (hI : ∀ (a : A), a ∈ I → f a = 0) :
A ⧸ I →ₐ[R₁] B | { commutes' := λ r, begin
-- this is is_scalar_tower.algebra_map_apply R₁ A (A ⧸ I) but the file `algebra.algebra.tower`
-- imports this file.
have : algebra_map R₁ (A ⧸ I) r = algebra_map A (A ⧸ I) (algebra_map R₁ A r),
{ simp_rw [algebra.algebra_map_eq_smul_one, smul_assoc, one_smul] },
rw [this, ... | def | ideal.quotient.liftₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.coe_to_ring_hom",
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"ideal",
"ideal.quotient.algebra_map_eq",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"map_one",
"one_smul",
"ring_hom.to_fun_eq_coe",
"smul_assoc"
] | `ideal.quotient.lift` as an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient.liftₐ_apply (I : ideal A) (f : A →ₐ[R₁] B) (hI : ∀ (a : A), a ∈ I → f a = 0) (x) :
ideal.quotient.liftₐ I f hI x = ideal.quotient.lift I (f : A →+* B) hI x | rfl | lemma | ideal.quotient.liftₐ_apply | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.lift",
"ideal.quotient.liftₐ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.liftₐ_comp (I : ideal A) (f : A →ₐ[R₁] B) (hI : ∀ (a : A), a ∈ I → f a = 0) :
(ideal.quotient.liftₐ I f hI).comp (ideal.quotient.mkₐ R₁ I) = f | alg_hom.ext (λ x, (ideal.quotient.lift_mk I (f : A →+* B) hI : _)) | lemma | ideal.quotient.liftₐ_comp | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.ext",
"ideal",
"ideal.quotient.lift_mk",
"ideal.quotient.liftₐ",
"ideal.quotient.mkₐ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift.map_smul (f : A →ₐ[R₁] B) (r : R₁) (x : A ⧸ f.to_ring_hom.ker) :
f.to_ring_hom.ker_lift (r • x) = r • f.to_ring_hom.ker_lift x | begin
obtain ⟨a, rfl⟩ := quotient.mkₐ_surjective R₁ _ x,
rw [← alg_hom.map_smul, quotient.mkₐ_eq_mk, ring_hom.ker_lift_mk],
exact f.map_smul _ _
end | lemma | ideal.ker_lift.map_smul | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.map_smul",
"ring_hom.ker_lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift_alg (f : A →ₐ[R₁] B) : (A ⧸ f.to_ring_hom.ker) →ₐ[R₁] B | alg_hom.mk' f.to_ring_hom.ker_lift (λ _ _, ker_lift.map_smul f _ _) | def | ideal.ker_lift_alg | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.mk'"
] | The induced algebras morphism from the quotient by the kernel to the codomain.
This is an isomorphism if `f` has a right inverse (`quotient_ker_alg_equiv_of_right_inverse`) /
is surjective (`quotient_ker_alg_equiv_of_surjective`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_lift_alg_mk (f : A →ₐ[R₁] B) (a : A) :
ker_lift_alg f (quotient.mk f.to_ring_hom.ker a) = f a | rfl | lemma | ideal.ker_lift_alg_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift_alg_to_ring_hom (f : A →ₐ[R₁] B) :
(ker_lift_alg f).to_ring_hom = ring_hom.ker_lift f | rfl | lemma | ideal.ker_lift_alg_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.ker_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_lift_alg_injective (f : A →ₐ[R₁] B) : function.injective (ker_lift_alg f) | ring_hom.ker_lift_injective f | lemma | ideal.ker_lift_alg_injective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.ker_lift_injective"
] | The induced algebra morphism from the quotient by the kernel is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_ker_alg_equiv_of_right_inverse
{f : A →ₐ[R₁] B} {g : B → A} (hf : function.right_inverse g f) :
(A ⧸ f.to_ring_hom.ker) ≃ₐ[R₁] B | { ..ring_hom.quotient_ker_equiv_of_right_inverse (λ x, show f.to_ring_hom (g x) = x, from hf x),
..ker_lift_alg f} | def | ideal.quotient_ker_alg_equiv_of_right_inverse | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.quotient_ker_equiv_of_right_inverse"
] | The **first isomorphism** theorem for algebras, computable version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_ker_alg_equiv_of_right_inverse.apply {f : A →ₐ[R₁] B} {g : B → A}
(hf : function.right_inverse g f) (x : A ⧸ f.to_ring_hom.ker) :
quotient_ker_alg_equiv_of_right_inverse hf x = ker_lift_alg f x | rfl | lemma | ideal.quotient_ker_alg_equiv_of_right_inverse.apply | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ker_alg_equiv_of_right_inverse_symm.apply {f : A →ₐ[R₁] B} {g : B → A}
(hf : function.right_inverse g f) (x : B) :
(quotient_ker_alg_equiv_of_right_inverse hf).symm x = quotient.mkₐ R₁ f.to_ring_hom.ker (g x) | rfl | lemma | ideal.quotient_ker_alg_equiv_of_right_inverse_symm.apply | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_ker_alg_equiv_of_surjective
{f : A →ₐ[R₁] B} (hf : function.surjective f) : (A ⧸ f.to_ring_hom.ker) ≃ₐ[R₁] B | quotient_ker_alg_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse) | def | ideal.quotient_ker_alg_equiv_of_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | The **first isomorphism theorem** for algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map {I : ideal R} (J : ideal S) (f : R →+* S) (hIJ : I ≤ J.comap f) :
R ⧸ I →+* S ⧸ J | (quotient.lift I ((quotient.mk J).comp f) (λ _ ha,
by simpa [function.comp_app, ring_hom.coe_comp, quotient.eq_zero_iff_mem] using hIJ ha)) | def | ideal.quotient_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"quotient_map",
"ring_hom.coe_comp"
] | The ring hom `R/I →+* S/J` induced by a ring hom `f : R →+* S` with `I ≤ f⁻¹(J)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_mk {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ I.comap f}
{x : R} : quotient_map I f H (quotient.mk J x) = quotient.mk I (f x) | quotient.lift_mk J _ _ | lemma | ideal.quotient_map_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"quotient.lift_mk",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_algebra_map {J : ideal A} {I : ideal S} {f : A →+* S} {H : J ≤ I.comap f}
{x : R₁} :
quotient_map I f H (algebra_map R₁ (A ⧸ J) x) = quotient.mk I (f (algebra_map _ _ x)) | quotient.lift_mk J _ _ | lemma | ideal.quotient_map_algebra_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ideal",
"quotient.lift_mk",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_comp_mk {J : ideal R} {I : ideal S} {f : R →+* S} (H : J ≤ I.comap f) :
(quotient_map I f H).comp (quotient.mk J) = (quotient.mk I).comp f | ring_hom.ext (λ x, by simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient_map_mk]) | lemma | ideal.quotient_map_comp_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient_map_mk",
"quotient_map",
"ring_hom.coe_comp",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = I.map (f : R →+* S)) :
R ⧸ I ≃+* S ⧸ J | { inv_fun := quotient_map I ↑f.symm (by {rw hIJ, exact le_of_eq (map_comap_of_equiv I f)}),
left_inv := by {rintro ⟨r⟩, simp },
right_inv := by {rintro ⟨s⟩, simp },
..quotient_map J ↑f (by {rw hIJ, exact @le_comap_map _ S _ _ _ _ _ _}) } | def | ideal.quotient_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"inv_fun",
"quotient_map"
] | The ring equiv `R/I ≃+* S/J` induced by a ring equiv `f : R ≃+** S`, where `J = f(I)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_equiv_mk (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = I.map (f : R →+* S))
(x : R) : quotient_equiv I J f hIJ (ideal.quotient.mk I x) = ideal.quotient.mk J (f x) | rfl | lemma | ideal.quotient_equiv_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_symm_mk (I : ideal R) (J : ideal S) (f : R ≃+* S)
(hIJ : J = I.map (f : R →+* S)) (x : S) :
(quotient_equiv I J f hIJ).symm (ideal.quotient.mk J x) = ideal.quotient.mk I (f.symm x) | rfl | lemma | ideal.quotient_equiv_symm_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_injective' {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ I.comap f}
(h : I.comap f ≤ J) : function.injective (quotient_map I f H) | begin
refine (injective_iff_map_eq_zero (quotient_map I f H)).2 (λ a ha, _),
obtain ⟨r, rfl⟩ := quotient.mk_surjective a,
rw [quotient_map_mk, quotient.eq_zero_iff_mem] at ha,
exact (quotient.eq_zero_iff_mem).mpr (h ha),
end | lemma | ideal.quotient_map_injective' | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"quotient_map"
] | `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_injective {I : ideal S} {f : R →+* S} :
function.injective (quotient_map I f le_rfl) | quotient_map_injective' le_rfl | lemma | ideal.quotient_map_injective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"le_rfl",
"quotient_map"
] | If we take `J = I.comap f` then `quotient_map` is injective automatically. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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