Loogle!

Result

Found 306 declarations mentioning RingHom.ker. Of these, only the first 200 are shown.

• RingHom.ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : Ideal R
• RingHom.instIsTwoSidedKer 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : (RingHom.ker f).IsTwoSided
• RingHom.ker_isPrime 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [IsDomain S] [FunLike F R S] [RingHomClass F R S] (f : F) : (RingHom.ker f).IsPrime
• RingHom.ker_eq_top_of_subsingleton 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Subsingleton S] (f : F) : RingHom.ker f = ⊤
• RingHom.ker_ne_top 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F) : RingHom.ker f ≠ ⊤
• RingHom.ker_eq_comap_bot 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : RingHom.ker f = Ideal.comap f ⊥
• RingHom.ker_isMaximal_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {K : Type u_2} {F : Type u_3} [Ring R] [DivisionRing K] [FunLike F R K] [RingHomClass F R K] (f : F) (hf : Function.Surjective ⇑f) : (RingHom.ker f).IsMaximal
• Ideal.ker_le_comap 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {K : Ideal S} (f : F) : RingHom.ker f ≤ Ideal.comap f K
• RingHom.one_notMem_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F) : 1 ∉ RingHom.ker f
• RingHom.injective_iff_ker_eq_bot 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) : Function.Injective ⇑f ↔ RingHom.ker f = ⊥
• RingHom.mem_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] {f : F} {r : R} : r ∈ RingHom.ker f ↔ f r = 0
• RingHom.ker_eq 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : ↑(RingHom.ker f) = ⇑f ⁻¹' {0}
• RingHom.ker_equiv 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] {F' : Type u_3} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') : RingHom.ker f = ⊥
• RingHom.ker_coe_toRingHom 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : RingHom.ker ↑f = RingHom.ker f
• Ideal.map_eq_bot_iff_le_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {I : Ideal R} (f : F) : Ideal.map f I = ⊥ ↔ I ≤ RingHom.ker f
• FaithfulSMul.ker_algebraMap_eq_bot 📋 Mathlib.RingTheory.Ideal.Maps
  (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥
• RingHom.ker_eq_bot_iff_eq_zero 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) : RingHom.ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0
• RingHom.sub_mem_ker_iff 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {x y : R} : x - y ∈ RingHom.ker f ↔ f x = f y
• Ideal.map_isPrime_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) {I : Ideal R} [H : I.IsPrime] (hk : RingHom.ker f ≤ I) : (Ideal.map f I).IsPrime
• Ideal.IsMaximal.map_of_surjective_of_ker_le 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) {m : Ideal R} [m.IsMaximal] (hk : RingHom.ker f ≤ m) : (Ideal.map f m).IsMaximal
• Ideal.comap_map_of_surjective' 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal R) : Ideal.comap f (Ideal.map f I) = I ⊔ RingHom.ker f
• RingHom.ker_equiv_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : R →+* S) (e : S ≃+* T) : RingHom.ker (e.toRingHom.comp f) = RingHom.ker f
• RingHom.comap_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* R) (g : T →+* S) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
• RingHom.ker_comp_of_injective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Ring R] [Semiring S] [Semiring T] (g : T →+* R) {f : R →+* S} (hf : Function.Injective ⇑f) : RingHom.ker (f.comp g) = RingHom.ker g
• Pi.ker_ringHom 📋 Mathlib.RingTheory.Ideal.Maps
  {S : Type v} [Semiring S] {ι : Type u_3} {R : ι → Type u_4} [(i : ι) → Semiring (R i)] (φ : (i : ι) → S →+* R i) : RingHom.ker (RingHom.pi φ) = ⨅ i, RingHom.ker (φ i)
• AlgHom.ker_coe_equiv 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) : RingHom.ker ↑e = ⊥
• AlgHom.ker_coe 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : RingHom.ker f = RingHom.ker ↑f
• Ideal.map_sInf 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] {A : Set (Ideal R)} {f : F} (hf : Function.Surjective ⇑f) : (∀ J ∈ A, RingHom.ker f ≤ J) → Ideal.map f (sInf A) = sInf (Ideal.map f '' A)
• RingHom.liftOfRightInverseAux 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) : B →+* C
• RingHom.liftOfSurjective 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) : { g // RingHom.ker f ≤ RingHom.ker g } ≃ (B →+* C)
• RingHom.liftOfRightInverse 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) : { g // RingHom.ker f ≤ RingHom.ker g } ≃ (B →+* C)
• RingHom.ker_coe_equiv 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R ≃+* S) : RingHom.ker ↑f = ⊥
• AlgHom.comap_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] (f : B →ₐ[R] C) (g : A →ₐ[R] B) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
• RingHom.ker_rangeSRestrict 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S) : RingHom.ker f.rangeSRestrict = RingHom.ker f
• RingHom.ker_evalRingHom 📋 Mathlib.RingTheory.Ideal.Maps
  {ι : Type u_1} [DecidableEq ι] (R : ι → Type u_2) [(i : ι) → CommRing (R i)] (i : ι) : RingHom.ker (Pi.evalRingHom R i) = Ideal.span {1 - Pi.single i 1}
• Ideal.map_radical_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : Function.Surjective ⇑f) {I : Ideal R} (h : RingHom.ker f ≤ I) : Ideal.map f I.radical = (Ideal.map f I).radical
• RingHom.liftOfRightInverseAux_comp_apply 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (a : A) : (f.liftOfRightInverseAux f_inv hf g hg) (f a) = g a
• Ideal.eq_map_of_comap_eq_ker_sup 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) (surj : Function.Surjective ⇑f) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap f I = RingHom.ker f ⊔ J) : I = Ideal.map f J
• Ideal.exists_of_comap_eq_ker_sup 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} [Ring A] [Ring B] (f : A →+* B) (surj : Function.Surjective ⇑f) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap f I = RingHom.ker f ⊔ J) {x : B} (hx : x ∈ I) : ∃ y ∈ J, f y = x
• Ideal.map_eq_iff_sup_ker_eq_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {I J : Ideal R} (f : R →+* S) (hf : Function.Surjective ⇑f) : Ideal.map f I = Ideal.map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker f
• RingHom.ker_rangeRestrict 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : RingHom.ker f.rangeRestrict = RingHom.ker f
• RingHom.liftOfSurjective_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) : ((f.liftOfSurjective hf) g).comp f = ↑g
• RingHom.liftOfRightInverse_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) : ((f.liftOfRightInverse f_inv hf) g).comp f = ↑g
• RingHom.eq_liftOfSurjective 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) : h = (f.liftOfSurjective hf) ⟨g, hg⟩
• RingHom.eq_liftOfRightInverse 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) : h = (f.liftOfRightInverse f_inv hf) ⟨g, hg⟩
• RingHom.liftOfSurjective_comp_apply 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) (x : A) : ((f.liftOfSurjective hf) g) (f x) = ↑g x
• RingHom.liftOfRightInverse_comp_apply 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) (x : A) : ((f.liftOfRightInverse f_inv hf) g) (f x) = ↑g x
• RingHom.ker_prodMap 📋 Mathlib.RingTheory.Ideal.Prod
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_1} {U : Type u_2} [Semiring T] [Semiring U] (f : R →+* S) (g : T →+* U) : RingHom.ker (f.prodMap g) = (RingHom.ker f).prod (RingHom.ker g)
• Polynomial.ker_mapRingHom 📋 Mathlib.RingTheory.Polynomial.Basic
  {R : Type u} {S : Type u_1} [CommSemiring R] [Semiring S] (f : R →+* S) : RingHom.ker (Polynomial.mapRingHom f) = Ideal.map Polynomial.C (RingHom.ker f)
• MvPolynomial.ker_map 📋 Mathlib.RingTheory.Polynomial.Basic
  {R : Type u} {S : Type u_1} {σ : Type v} [CommRing R] [CommRing S] (f : R →+* S) : RingHom.ker (MvPolynomial.map f) = Ideal.map MvPolynomial.C (RingHom.ker f)
• MvPolynomial.ker_mapAlgHom 📋 Mathlib.RingTheory.Polynomial.Basic
  {R : Type u} [CommRing R] {S₁ : Type u_2} {S₂ : Type u_3} {σ : Type u_4} [CommRing S₁] [CommRing S₂] [Algebra R S₁] [Algebra R S₂] (f : S₁ →ₐ[R] S₂) : RingHom.ker (MvPolynomial.mapAlgHom f) = Ideal.map MvPolynomial.C (RingHom.ker f)
• AlgHom.ker_rangeRestrict 📋 Mathlib.Algebra.Algebra.Subalgebra.Operations
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : RingHom.ker f.rangeRestrict = RingHom.ker f
• Ideal.mk_ker 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] : RingHom.ker (Ideal.Quotient.mk I) = I
• RingHom.kerLift 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) : R ⧸ RingHom.ker f →+* S
• Ideal.injective_lift_iff 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {I : Ideal R} [I.IsTwoSided] {f : R →+* S} (H : ∀ a ∈ I, f a = 0) : Function.Injective ⇑(Ideal.Quotient.lift I f H) ↔ RingHom.ker f = I
• RingHom.lift_injective_of_ker_le_ideal 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] (I : Ideal R) [I.IsTwoSided] {f : R →+* S} (H : ∀ a ∈ I, f a = 0) (hI : RingHom.ker f ≤ I) : Function.Injective ⇑(Ideal.Quotient.lift I f H)
• AlgHom.liftOfSurjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) : B →ₐ[R] C
• Ideal.ker_Pi_Quotient_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] : RingHom.ker (RingHom.pi fun i => Ideal.Quotient.mk (I i)) = ⨅ i, I i
• AlgHom.liftOfSurjective_comp 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) : (f.liftOfSurjective hf g H).comp f = g
• AlgHom.liftOfSurjective_surjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) (hg : Function.Surjective ⇑g) : Function.Surjective ⇑(f.liftOfSurjective hf g H)
• Ideal.ker_quotient_lift 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {I : Ideal R} [I.IsTwoSided] (f : R →+* S) (H : I ≤ RingHom.ker f) : RingHom.ker (Ideal.Quotient.lift I f H) = Ideal.map (Ideal.Quotient.mk I) (RingHom.ker f)
• AlgHom.liftOfSurjective_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) (x : A) : (f.liftOfSurjective hf g H) (f x) = g x
• Ideal.kerLiftAlg 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) : A ⧸ RingHom.ker f →ₐ[R₁] B
• Ideal.Quotient.mkₐ_ker 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] : RingHom.ker ↑(Ideal.Quotient.mkₐ R₁ I) = I
• Ideal.quotientKerAlgEquivOfSurjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) : (A ⧸ RingHom.ker f) ≃ₐ[R₁] B
• Ideal.quotientKerAlgEquivOfRightInverse 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) : (A ⧸ RingHom.ker f) ≃ₐ[R₁] B
• Ideal.quotientKerEquivRange 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_5} {A : Type u_6} {B : Type u_7} [CommSemiring R] [Ring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : (A ⧸ RingHom.ker f) ≃ₐ[R] ↥f.range
• RingHom.quotientKerEquivOfSurjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) : R ⧸ RingHom.ker f ≃+* S
• RingHom.quotientKerEquivOfRightInverse 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) : R ⧸ RingHom.ker f ≃+* S
• RingHom.kerLift_injective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) : Function.Injective ⇑f.kerLift
• RingHom.quotientKerEquivRangeS 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) : R ⧸ RingHom.ker f ≃+* ↥f.rangeS
• DoubleQuot.ker_quotLeftToQuotSup 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [CommRing R] (I J : Ideal R) : RingHom.ker (DoubleQuot.quotLeftToQuotSup I J) = Ideal.map (Ideal.Quotient.mk I) J
• RingHom.quotientKerEquivRange 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R →+* S) : R ⧸ RingHom.ker f ≃+* ↥f.range
• Ideal.kerLiftAlg_injective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) : Function.Injective ⇑(Ideal.kerLiftAlg f)
• RingHom.kerLift_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) (r : R) : f.kerLift ((Ideal.Quotient.mk (RingHom.ker f)) r) = f r
• RingHom.quotientKerEquivOfSurjective_symm_comp 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) : (RingHom.quotientKerEquivOfSurjective hf).symm.toRingHom.comp f = Ideal.Quotient.mk (RingHom.ker f)
• Ideal.KerLift.map_smul 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) (r : R₁) (x : A ⧸ RingHom.ker f) : f.kerLift (r • x) = r • f.kerLift x
• Ideal.quotientKerAlgEquivOfSurjective_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A ⧸ RingHom.ker f.toRingHom) : (Ideal.quotientKerAlgEquivOfSurjective hf) a = (↑f).kerLift a
• Ideal.quotientKerAlgEquivOfRightInverse_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) (a : A ⧸ RingHom.ker f.toRingHom) : (Ideal.quotientKerAlgEquivOfRightInverse hf) a = (↑f).kerLift a
• Ideal.kerLiftAlg_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) (a : A) : (Ideal.kerLiftAlg f) ((Ideal.Quotient.mk (RingHom.ker f)) a) = f a
• Ideal.quotientKerAlgEquivOfSurjective_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A) : (Ideal.quotientKerAlgEquivOfSurjective hf) ((Ideal.Quotient.mk (RingHom.ker f)) a) = f a
• DoubleQuot.ker_quotQuotMk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [CommRing R] (I J : Ideal R) : RingHom.ker (DoubleQuot.quotQuotMk I J) = I ⊔ J
• Ideal.quotientKerAlgEquivOfRightInverse_symm_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) (a✝ : B) : (Ideal.quotientKerAlgEquivOfRightInverse hf).symm a✝ = (Ideal.Quotient.mk (RingHom.ker ↑f)) (g a✝)
• RingHom.quotientKerEquivOfRightInverse.apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) (x : R ⧸ RingHom.ker f) : (RingHom.quotientKerEquivOfRightInverse hf) x = f.kerLift x
• RingHom.quotientKerEquivOfSurjective_apply_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (x : R) : (RingHom.quotientKerEquivOfSurjective hf) ((Ideal.Quotient.mk (RingHom.ker f)) x) = f x
• Ideal.quotientKerAlgEquivOfSurjective_symm_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A) : (Ideal.quotientKerAlgEquivOfSurjective hf).symm (f a) = (Ideal.Quotient.mk (RingHom.ker f)) a
• RingHom.quotientKerEquivOfRightInverse.Symm.apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) (x : S) : (RingHom.quotientKerEquivOfRightInverse hf).symm x = (Ideal.Quotient.mk (RingHom.ker f)) (g x)
• RingHom.quotientKerEquivOfSurjective_symm_apply 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (x : R) : (RingHom.quotientKerEquivOfSurjective hf).symm (f x) = (Ideal.Quotient.mk (RingHom.ker f)) x
• Ideal.ker_quotientMap_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] : RingHom.ker (Ideal.quotientMap (Ideal.map (Ideal.Quotient.mk I) J) (Ideal.Quotient.mk I) ⋯) = Ideal.map (Ideal.Quotient.mk J) I
• Ideal.kerLiftAlg_toRingHom 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) : ↑(Ideal.kerLiftAlg f) = (↑f).kerLift
• transcendental_iff_ker_eq_bot 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] {x : A} : Transcendental R x ↔ RingHom.ker (Polynomial.aeval x) = ⊥
• Algebra.TensorProduct.rTensor_ker 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_4} {S : Type u_5} [CommRing R] [CommRing S] [Algebra R S] {A : Type u_6} {B : Type u_7} {C : Type u_8} [Ring A] [Ring B] [Ring C] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra S A] [Algebra S B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) (hf : Function.Surjective ⇑f) : RingHom.ker (Algebra.TensorProduct.map f (AlgHom.id R C)) = Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f)
• Algebra.TensorProduct.lTensor_ker 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_4} [CommRing R] {A : Type u_6} {C : Type u_8} {D : Type u_9} [Ring A] [Ring C] [Ring D] [Algebra R A] [Algebra R C] [Algebra R D] (g : C →ₐ[R] D) (hg : Function.Surjective ⇑g) : RingHom.ker (Algebra.TensorProduct.map (AlgHom.id R A) g) = Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g)
• Algebra.TensorProduct.map_ker 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_4} {S : Type u_5} [CommRing R] [CommRing S] [Algebra R S] {A : Type u_6} {B : Type u_7} {C : Type u_8} {D : Type u_9} [Ring A] [Ring B] [Ring C] [Ring D] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D] [Algebra S A] [Algebra S B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) (g : C →ₐ[R] D) (hf : Function.Surjective ⇑f) (hg : Function.Surjective ⇑g) : RingHom.ker (Algebra.TensorProduct.map f g) = Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f) ⊔ Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g)
• IsLocalization.away_of_isIdempotentElem 📋 Mathlib.RingTheory.Localization.Away.Basic
  {R : Type u_4} {S : Type u_5} [CommRing R] [CommRing S] [Algebra R S] {e : R} (he : IsIdempotentElem e) (H : RingHom.ker (algebraMap R S) = Ideal.span {1 - e}) (H' : Function.Surjective ⇑(algebraMap R S)) : IsLocalization.Away e S
• IsLocalization.Away.of_surjective 📋 Mathlib.RingTheory.Localization.Away.Basic
  {S : Type u_4} {T : Type u_5} [CommRing S] [CommRing T] [Algebra S T] (h₁ : Function.Surjective ⇑(algebraMap S T)) {r : S} (hr : IsUnit ((algebraMap S T) r)) (n : ℕ) (hn : r ^ n • RingHom.ker (algebraMap S T) ≤ ⊥) : IsLocalization.Away r T
• IsLocalization.Away.of_surjective_of_isScalarTower 📋 Mathlib.RingTheory.Localization.Away.Basic
  {S : Type u_4} {T : Type u_5} [CommRing S] [CommRing T] [Algebra S T] {R : Type u_6} [CommRing R] [Algebra R S] [Algebra R T] [IsScalarTower R S T] (h₁ : Function.Surjective ⇑(algebraMap S T)) (h₂ : Function.Surjective ⇑(algebraMap R S)) (r : R) (hr : IsUnit ((algebraMap R T) r)) {n : ℕ} (hn : r ^ n • RingHom.ker (algebraMap R T) ≤ RingHom.ker (algebraMap R S)) : IsLocalization.Away ((algebraMap R S) r) T
• Ring.map_jacobson_of_ker_le 📋 Mathlib.RingTheory.Jacobson.Radical
  {R : Type u_1} {R₂ : Type u_2} [Ring R] [Ring R₂] {f : R →+* R₂} [RingHomSurjective f] (le : RingHom.ker f ≤ Ring.jacobson R) : Submodule.map f.toSemilinearMap (Ring.jacobson R) = Ring.jacobson R₂
• Ideal.map_jacobson_of_surjective 📋 Mathlib.RingTheory.Jacobson.Ideal
  {R : Type u} {S : Type v} [Ring R] [Ring S] {I : Ideal R} {f : R →+* S} (hf : Function.Surjective ⇑f) : RingHom.ker f ≤ I → Ideal.map f I.jacobson = (Ideal.map f I).jacobson
• IsLocalRing.ker_eq_maximalIdeal 📋 Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
  {R : Type u_1} {K : Type u_2} [CommRing R] [IsLocalRing R] [DivisionRing K] (φ : R →+* K) (hφ : Function.Surjective ⇑φ) : RingHom.ker φ = IsLocalRing.maximalIdeal R
• Ideal.fg_of_fg_map_of_fg_inf_ker_of_surjective 📋 Mathlib.RingTheory.Finiteness.Ideal
  {R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} {I : Ideal R} (hmap : (Ideal.map f I).FG) (hk : (I ⊓ RingHom.ker f).FG) (hf : Function.Surjective ⇑f) : I.FG
• Ideal.fg_ker_comp 📋 Mathlib.RingTheory.Finiteness.Ideal
  {R : Type u_3} {S : Type u_4} {A : Type u_5} [CommRing R] [CommRing S] [CommRing A] (f : R →+* S) (g : S →+* A) (hf : (RingHom.ker f).FG) (hg : (RingHom.ker g).FG) (hsur : Function.Surjective ⇑f) : (RingHom.ker (g.comp f)).FG
• RingHom.FinitePresentation.of_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) (hf : Function.Surjective ⇑f) (hker : (RingHom.ker f).FG) : f.FinitePresentation
• RingHom.FinitePresentation.comp_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (hf : f.FinitePresentation) (hg : Function.Surjective ⇑g) (hker : (RingHom.ker g).FG) : (g.comp f).FinitePresentation
• AlgHom.FinitePresentation.of_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (hker : (RingHom.ker f.toRingHom).FG) : f.FinitePresentation
• Algebra.FinitePresentation.ker_fG_of_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} {B : Type w₃} [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) [Algebra.FinitePresentation R A] [Algebra.FinitePresentation R B] : (RingHom.ker f.toRingHom).FG
• Algebra.FinitePresentation.of_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} {B : Type w₃} [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] {f : A →ₐ[R] B} (hf : Function.Surjective ⇑f) (hker : (RingHom.ker f.toRingHom).FG) [Algebra.FinitePresentation R A] : Algebra.FinitePresentation R B
• AlgHom.FinitePresentation.comp_surjective 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.FinitePresentation) (hg : Function.Surjective ⇑g) (hker : (RingHom.ker g.toRingHom).FG) : (g.comp f).FinitePresentation
• RingHom.FinitePresentation.polynomial_induction 📋 Mathlib.RingTheory.FinitePresentation
  (P : (R : Type u) → [inst : CommRing R] → (S : Type u) → [inst_1 : CommRing S] → (R →+* S) → Prop) (Q : (R : Type u) → [inst : CommRing R] → (S : Type v) → [inst_1 : CommRing S] → (R →+* S) → Prop) (polynomial : ∀ (R : Type u) [inst : CommRing R], P R (Polynomial R) Polynomial.C) (fg_ker : ∀ (R : Type u) [inst : CommRing R] (S : Type v) [inst_1 : CommRing S] (f : R →+* S), Function.Surjective ⇑f → (RingHom.ker f).FG → Q R S f) (comp : ∀ (R : Type u) [inst : CommRing R] (S : Type u) [inst_1 : CommRing S] (T : Type v) [inst_2 : CommRing T] (f : R →+* S) (g : S →+* T), P R S f → Q S T g → Q R T (g.comp f)) {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.FinitePresentation) : Q R S f
• Algebra.FinitePresentation.mk 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} [CommSemiring R] [Semiring A] [Algebra R A] (out : ∃ n f, Function.Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG) : Algebra.FinitePresentation R A
• Algebra.FinitePresentation.out 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} {inst✝² : Algebra R A} [self : Algebra.FinitePresentation R A] : ∃ n f, Function.Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
• Algebra.FinitePresentation.ker_fg_of_mvPolynomial 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} [CommRing R] [CommRing A] [Algebra R A] {n : ℕ} (f : MvPolynomial (Fin n) R →ₐ[R] A) (hf : Function.Surjective ⇑f) [Algebra.FinitePresentation R A] : (RingHom.ker f.toRingHom).FG
• Algebra.FinitePresentation.iff_quotient_mvPolynomial' 📋 Mathlib.RingTheory.FinitePresentation
  {R : Type w₁} {A : Type w₂} [CommRing R] [CommRing A] [Algebra R A] : Algebra.FinitePresentation R A ↔ ∃ ι x f, Function.Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
• IsLocalization.of_surjective 📋 Mathlib.RingTheory.Localization.Ideal
  {R : Type u_1} [CommRing R] (M : Submonoid R) (S : Type u_2) [CommRing S] [Algebra R S] [IsLocalization M S] {R' : Type u_3} {S' : Type u_4} [CommRing R'] [CommRing S'] [Algebra R' S'] (f : R →+* R') (hf : Function.Surjective ⇑f) (g : S →+* S') (hg : Function.Surjective ⇑g) (H : g.comp (algebraMap R S) = (algebraMap R' S').comp f) (H' : RingHom.ker g ≤ Ideal.map (algebraMap R S) (RingHom.ker f)) : IsLocalization (Submonoid.map f M) S'
• RingHom.ker_isRadical_iff_reduced_of_surjective 📋 Mathlib.RingTheory.Nilpotent.Lemmas
  {R : Type u_1} {S : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring S] [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective ⇑f) : (RingHom.ker f).IsRadical ↔ IsReduced S
• RingHom.toKerIsLocalization 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_1} (S : Type u_2) {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hy : M ≤ Submonoid.comap g T) : ↥(RingHom.ker g) →ₗ[R] ↥(RingHom.ker (IsLocalization.map Q g hy))
• AlgHom.toKerIsLocalization 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_5} [CommSemiring R] (M : Submonoid R) {A : Type u_6} [CommSemiring A] [Algebra R A] {B : Type u_7} [CommSemiring B] [Algebra R B] (Rₚ : Type u_8) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type u_9) [CommSemiring Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type u_10) [CommSemiring Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) : ↥(RingHom.ker f) →ₗ[A] ↥(RingHom.ker (IsLocalization.mapₐ M Rₚ Aₚ Bₚ f))
• AlgHom.toKerIsLocalization_isLocalizedModule 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_5} [CommSemiring R] (M : Submonoid R) {A : Type u_6} [CommSemiring A] [Algebra R A] {B : Type u_7} [CommSemiring B] [Algebra R B] (Rₚ : Type u_8) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type u_9) [CommSemiring Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type u_10) [CommSemiring Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) : IsLocalizedModule (Algebra.algebraMapSubmonoid A M) (AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f)
• IsLocalization.ker_map 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hT : Submonoid.map g M = T) : RingHom.ker (IsLocalization.map Q g ⋯) = Ideal.map (algebraMap R S) (RingHom.ker g)
• RingHom.toKerIsLocalization_apply 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hy : M ≤ Submonoid.comap g T) (r : ↥(RingHom.ker g)) : ↑((RingHom.toKerIsLocalization S Q g hy) r) = (algebraMap R S) ↑r
• AlgHom.toKerIsLocalization_apply 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_5} [CommSemiring R] (M : Submonoid R) {A : Type u_6} [CommSemiring A] [Algebra R A] {B : Type u_7} [CommSemiring B] [Algebra R B] (Rₚ : Type u_8) [CommSemiring Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type u_9) [CommSemiring Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type u_10) [CommSemiring Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) (x : ↥(RingHom.ker f)) : (AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f) x = (RingHom.toKerIsLocalization Aₚ Bₚ f.toRingHom ⋯) x
• RingHom.toKerIsLocalization_isLocalizedModule 📋 Mathlib.RingTheory.Localization.Algebra
  {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hT : Submonoid.map g M = T) : IsLocalizedModule M (RingHom.toKerIsLocalization S Q g ⋯)
• Ideal.iInf_ker_le 📋 Mathlib.RingTheory.LocalProperties.Basic
  {R : Type u_1} [CommSemiring R] (I : Ideal R) : ⨅ p, ⨅ (x : p.IsPrime), ⨅ (_ : I ≤ p), RingHom.ker (algebraMap R (Localization.AtPrime p)) ≤ I
• Algebra.ker_algebraMap_isMaximal_of_isIntegral 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
  (R : Type u_1) [CommRing R] (k : Type u_6) [Field k] [Algebra R k] [Algebra.IsIntegral R k] : (RingHom.ker (algebraMap R k)).IsMaximal
• RingHom.IsIntegral.kerLift 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
  {S : Type u_4} {T : Type u_5} [CommRing S] [CommRing T] {f : S →+* T} (hf : f.IsIntegral) : f.kerLift.IsIntegral
• Ideal.exists_ideal_over_maximal_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P_max : P.IsMaximal] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsMaximal ∧ Ideal.comap (algebraMap R S) Q = P
• Ideal.map_eq_top_iff_of_ker_le 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal R} (hf₁ : RingHom.ker f ≤ I) (hf₂ : f.IsIntegral) : Ideal.map f I = ⊤ ↔ I = ⊤
• Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain S] [Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = P
• IsLocalRing.ker_residue 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} [CommRing R] [IsLocalRing R] : RingHom.ker (IsLocalRing.residue R) = IsLocalRing.maximalIdeal R
• Ideal.ResidueField.lift 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (I : Ideal R) [I.IsPrime] (f : R →+* S) (hf₁ : I ≤ RingHom.ker f) (hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid S)) : I.ResidueField →+* S
• Ideal.ResidueField.liftₐ 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal A) [I.IsPrime] (f : A →ₐ[R] B) (hf₁ : I ≤ RingHom.ker f) (hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid B)) : I.ResidueField →ₐ[R] B
• Ideal.ker_algebraMap_residueField 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} [CommRing R] (I : Ideal R) [I.IsPrime] : RingHom.ker (algebraMap R I.ResidueField) = I
• Ideal.ResidueField.lift_algebraMap 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (I : Ideal R) [I.IsPrime] (f : R →+* S) (hf₁ : I ≤ RingHom.ker f) (hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid S)) (r : R) : (Ideal.ResidueField.lift I f hf₁ hf₂) ((algebraMap R I.ResidueField) r) = f r
• Ideal.ResidueField.liftₐ_comp_toAlgHom 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal A) [I.IsPrime] (f : A →ₐ[R] B) (hf₁ : I ≤ RingHom.ker f) (hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid B)) : (Ideal.ResidueField.liftₐ I f hf₁ hf₂).comp (IsScalarTower.toAlgHom R A I.ResidueField) = f
• Ideal.ResidueField.liftₐ_algebraMap 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
  {R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal A) [I.IsPrime] (f : A →ₐ[R] B) (hf₁ : I ≤ RingHom.ker f) (hf₂ : I.primeCompl ≤ Submonoid.comap f (IsUnit.submonoid B)) (r : A) : (Ideal.ResidueField.liftₐ I f hf₁ hf₂) ((algebraMap A I.ResidueField) r) = f r
• range_comap_of_surjective 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
  {R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) : Set.range (PrimeSpectrum.comap f) = PrimeSpectrum.zeroLocus ↑(RingHom.ker f)
• Ideal.primeSpectrumOrderIsoZeroLocusOfSurj 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) {I : Ideal R} (hI : RingHom.ker f = I) : PrimeSpectrum S ≃o ↑(PrimeSpectrum.zeroLocus ↑I)
• Ideal.minimalPrimes_map_of_surjective 📋 Mathlib.RingTheory.Ideal.MinimalPrime.Localization
  {R : Type u_1} [CommRing R] {S : Type u_3} [CommRing S] {f : R →+* S} (hf : Function.Surjective ⇑f) (I : Ideal R) : (Ideal.map f I).minimalPrimes = Ideal.map f '' (I ⊔ RingHom.ker f).minimalPrimes
• PrimeSpectrum.vanishingIdeal_range_comap 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : PrimeSpectrum.vanishingIdeal (Set.range (PrimeSpectrum.comap f)) = (RingHom.ker f).radical
• PrimeSpectrum.closure_range_comap 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : closure (Set.range (PrimeSpectrum.comap f)) = PrimeSpectrum.zeroLocus ↑(RingHom.ker f)
• PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) : DenseRange (PrimeSpectrum.comap f) ↔ RingHom.ker f ≤ nilradical R
• PrimeSpectrum.range_comap_fst 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] : Set.range (PrimeSpectrum.comap (RingHom.fst R S)) = PrimeSpectrum.zeroLocus ↑(RingHom.ker (RingHom.fst R S))
• PrimeSpectrum.range_comap_snd 📋 Mathlib.RingTheory.Spectrum.Prime.Topology
  {R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] : Set.range (PrimeSpectrum.comap (RingHom.snd R S)) = PrimeSpectrum.zeroLocus ↑(RingHom.ker (RingHom.snd R S))
• Module.AEval.annihilator_eq_ker_aeval 📋 Mathlib.Algebra.Polynomial.Module.AEval
  {R : Type u_3} {A : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [FaithfulSMul A M] : Module.annihilator (Polynomial R) (Module.AEval R M a) = RingHom.ker (Polynomial.aeval a)
• Module.AEval.annihilator_top_eq_ker_aeval 📋 Mathlib.Algebra.Polynomial.Module.AEval
  {R : Type u_3} {A : Type u_1} {M : Type u_2} [CommSemiring R] [Semiring A] (a : A) [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] [FaithfulSMul A M] : ⊤.annihilator = RingHom.ker (Polynomial.aeval a)
• Ideal.Algebra.kerSquareLift 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] {A : Type u_1} [CommRing A] [Algebra R A] : Algebra (R ⧸ RingHom.ker (algebraMap R A) ^ 2) A
• AlgHom.kerSquareLift 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : A ⧸ RingHom.ker f.toRingHom ^ 2 →ₐ[R] B
• Ideal.instIsScalarTowerQuotientHPowKerRingHomAlgebraMapOfNat 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] : IsScalarTower R (A ⧸ RingHom.ker (algebraMap A B) ^ 2) B
• Ideal.mapCotangent_surjective_of_comap_eq 📋 Mathlib.RingTheory.Ideal.Cotangent
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (surj : Function.Surjective ⇑(algebraMap A B)) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J) : Function.Surjective ⇑(J.mapCotangent I (Algebra.ofId A B) ⋯)
• Ideal.mapCotangent_ker_of_surjective 📋 Mathlib.RingTheory.Ideal.Cotangent
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (surj : Function.Surjective ⇑(algebraMap A B)) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J) : (J.mapCotangent I (Algebra.ofId A B) ⋯).ker = Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (RingHom.ker (algebraMap A B) ⊓ J))
• AlgHom.ker_kerSquareLift 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : RingHom.ker f.kerSquareLift.toRingHom = (RingHom.ker f.toRingHom).cotangentIdeal
• AlgHom.kerSquareLift_mk 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (x : A) : f.kerSquareLift ((Ideal.Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x) = f x
• KaehlerDifferential.kerToTensor 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] : ↥(RingHom.ker (algebraMap A B)) →ₗ[A] TensorProduct A B Ω[A⁄R]
• KaehlerDifferential.kerCotangentToTensor 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] : (RingHom.ker (algebraMap A B)).Cotangent →ₗ[A] TensorProduct A B Ω[A⁄R]
• KaehlerDifferential.range_kerCotangentToTensor 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (h : Function.Surjective ⇑(algebraMap A B)) : (KaehlerDifferential.kerCotangentToTensor R A B).range = Submodule.restrictScalars A (KaehlerDifferential.mapBaseChange R A B).ker
• KaehlerDifferential.kerToTensor_apply 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] (x : ↥(RingHom.ker (algebraMap A B))) : (KaehlerDifferential.kerToTensor R A B) x = 1 ⊗ₜ[A] (KaehlerDifferential.D R A) ↑x
• KaehlerDifferential.exact_kerCotangentToTensor_mapBaseChange 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (h : Function.Surjective ⇑(algebraMap A B)) : Function.Exact ⇑(KaehlerDifferential.kerCotangentToTensor R A B) ⇑(KaehlerDifferential.mapBaseChange R A B)
• KaehlerDifferential.kerCotangentToTensor_toCotangent 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra A B] (x : ↥(RingHom.ker (algebraMap A B))) : (KaehlerDifferential.kerCotangentToTensor R A B) ((RingHom.ker (algebraMap A B)).toCotangent x) = 1 ⊗ₜ[A] (KaehlerDifferential.D R A) ↑x
• KaehlerDifferential.endEquiv 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module.End S Ω[S⁄R] ≃ { f // (Algebra.TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R S }
• KaehlerDifferential.End_equiv_aux 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (f : S →ₐ[R] TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) : (Ideal.Quotient.mkₐ R (KaehlerDifferential.ideal R S).cotangentIdeal).comp f = IsScalarTower.toAlgHom R S ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ (KaehlerDifferential.ideal R S).cotangentIdeal) ↔ (Algebra.TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R S
• KaehlerDifferential.endEquivAuxEquiv 📋 Mathlib.RingTheory.Kaehler.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : { f // (Ideal.Quotient.mkₐ R (KaehlerDifferential.ideal R S).cotangentIdeal).comp f = IsScalarTower.toAlgHom R S ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ (KaehlerDifferential.ideal R S).cotangentIdeal) } ≃ { f // (Algebra.TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R S }
• Algebra.kerTensorProductMapIdToAlgHomEquiv 📋 Mathlib.RingTheory.Flat.Equalizer
  (R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (T : Type u_3) [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (A : Type u_4) [CommRing A] [Algebra R A] [Module.Flat R T] (h₁ : Function.Surjective ⇑(algebraMap S T)) : ↥(RingHom.ker (Algebra.TensorProduct.map (AlgHom.id A A) (IsScalarTower.toAlgHom R S T))) ≃ₗ[TensorProduct R A S] TensorProduct S (TensorProduct R A S) ↥(RingHom.ker (algebraMap S T))
• Algebra.kerTensorProductMapIdToAlgHomEquiv_symm_apply 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (T : Type u_3) [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {A : Type u_4} [CommRing A] [Algebra R A] [Module.Flat R T] (h₁ : Function.Surjective ⇑(algebraMap S T)) (x : A) (y : S) (z : ↥(RingHom.ker (algebraMap S T))) : ↑((Algebra.kerTensorProductMapIdToAlgHomEquiv R S T A h₁).symm (x ⊗ₜ[R] y ⊗ₜ[S] z)) = x ⊗ₜ[R] (y * ↑z)
• CharP.ker_intAlgebraMap_eq_span 📋 Mathlib.Algebra.CharP.Quotient
  {R : Type u_1} [Ring R] (p : ℕ) [CharP R p] : RingHom.ker (algebraMap ℤ R) = Ideal.span {↑p}
• CharP.quotient_iff_le_ker_natCast 📋 Mathlib.Algebra.CharP.Quotient
  {R : Type u_1} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) : CharP (R ⧸ I) n ↔ Ideal.comap (Nat.castRingHom R) I ≤ RingHom.ker (Nat.castRingHom R)
• CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {I : Type u_3} {e : I → R} [Fintype I] (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (he : ∀ (i : I), IsIdempotentElem (e i)) (he' : ∀ (i : I), IsMulCentral (e i)) (he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)) : CompleteOrthogonalIdempotents e
• exists_isIdempotentElem_eq_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e : S) (he : e ∈ f.range) (he' : IsIdempotentElem e) : ∃ e', IsIdempotentElem e' ∧ f e' = e
• OrthogonalIdempotents.lift_of_isNilpotent_ker 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {I : Type u_3} [Finite I] (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {e : I → S} (he : OrthogonalIdempotents e) (he' : ∀ (i : I), e i ∈ f.range) : ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {I : Type u_3} [Fintype I] (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {e : I → S} (he : CompleteOrthogonalIdempotents e) (he' : ∀ (i : I), e i ∈ f.range) : ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] (f : R →+* S) {I : Type u_3} [Fintype I] {e : I → R} (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (he : ∀ (i : I), IsIdempotentElem (e i)) (he' : CompleteOrthogonalIdempotents (⇑f ∘ e)) : CompleteOrthogonalIdempotents e
• OrthogonalIdempotents.lift_of_isNilpotent_ker_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {n : ℕ} {e : Fin n → S} (he : OrthogonalIdempotents e) (he' : ∀ (i : Fin n), e i ∈ f.range) : ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {n : ℕ} {e : Fin n → S} (he : CompleteOrthogonalIdempotents e) (he' : ∀ (i : Fin n), e i ∈ f.range) : ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.exists_eq_comp_of_ker_eq_span 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] {I : Type u_3} [Fintype I] (f : R →+* S) (e₀ : R) (he₀ : IsIdempotentElem e₀) (hfe₀ : RingHom.ker f = Ideal.span {e₀}) (e : I → S) (he : CompleteOrthogonalIdempotents e) (hef : ∀ (i : I), e i ∈ f.range) : ∃ e', (CompleteOrthogonalIdempotents fun t => Option.rec e₀ e' t) ∧ e = ⇑f ∘ e'
• existsUnique_isIdempotentElem_eq_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e : S) (he : e ∈ f.range) (he' : IsIdempotentElem e) : ∃! e', IsIdempotentElem e' ∧ f e' = e
• exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotent_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e₁ : S) (he : e₁ ∈ f.range) (he₁ : IsIdempotentElem e₁) (e₂ : R) (he₂ : IsIdempotentElem e₂) (he₁e₂ : e₁ * f e₂ = 0) : ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0
• exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e₁ : S) (he : e₁ ∈ f.range) (he₁ : IsIdempotentElem e₁) (e₂ : R) (he₂ : IsIdempotentElem e₂) (he₁e₂ : e₁ * f e₂ = 0) (he₂e₁ : f e₂ * e₁ = 0) : ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0 ∧ e₂ * e' = 0
• Polynomial.generalizedEisenstein 📋 Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
  {R : Type u_1} [CommRing R] [IsDomain R] {K : Type u_2} [Field K] [Algebra R K] {q f : Polynomial R} {p : ℕ} (hq_irr : Irreducible (Polynomial.map (algebraMap R K) q)) (hq_monic : q.Monic) (hf_prim : f.IsPrimitive) (hfd0 : 0 < f.natDegree) (hfP : (algebraMap R K) f.leadingCoeff ≠ 0) (hfmodP : Polynomial.map (algebraMap R K) f = Polynomial.C ((algebraMap R K) f.leadingCoeff) * Polynomial.map (algebraMap R K) q ^ p) (hfmodP2 : Polynomial.map (Ideal.Quotient.mk (RingHom.ker (algebraMap R K) ^ 2)) (f %ₘ q) ≠ 0) : Irreducible f
• minpoly.ker_aeval_eq_span_minpoly 📋 Mathlib.FieldTheory.Minpoly.Field
  (A : Type u_1) {B : Type u_2} [Field A] [Ring B] [Algebra A B] (x : B) : RingHom.ker (Polynomial.aeval x) = Polynomial A ∙ minpoly A x
• Polynomial.ker_constantCoeff 📋 Mathlib.RingTheory.Polynomial.Ideal
  {R : Type u_1} [CommRing R] : RingHom.ker Polynomial.constantCoeff = Ideal.span {Polynomial.X}
• Polynomial.ker_evalRingHom 📋 Mathlib.RingTheory.Polynomial.Ideal
  {R : Type u_1} [CommRing R] (x : R) : RingHom.ker (Polynomial.evalRingHom x) = Ideal.span {Polynomial.X - Polynomial.C x}
• Ideal.Quotient.factor_ker 📋 Mathlib.RingTheory.Ideal.Quotient.PowTransition
  {R : Type u_1} [Ring R] {I J : Ideal R} (H : I ≤ J) [I.IsTwoSided] [J.IsTwoSided] : RingHom.ker (Ideal.Quotient.factor H) = Ideal.map (Ideal.Quotient.mk I) J
• HomogeneousIdeal.toIdeal_irrelevant 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
  {ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] : (HomogeneousIdeal.irrelevant 𝒜).toIdeal = RingHom.ker (GradedRing.projZeroRingHom 𝒜)
• Algebra.Extension.Cotangent.map_surjective_of_comap_eq 📋 Mathlib.RingTheory.Extension.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {P' : Algebra.Extension R S} {f : P.Hom P'} (h : Function.Surjective ⇑f) (eq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker) : Function.Surjective ⇑(Algebra.Extension.Cotangent.map f)
• Algebra.Extension.Cotangent.map_ker_of_surjective 📋 Mathlib.RingTheory.Extension.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Extension R S} {P' : Algebra.Extension R S} {f : P.Hom P'} (h : Function.Surjective ⇑f) (eq : Ideal.comap f.toRingHom P'.ker = RingHom.ker f.toRingHom ⊔ P.ker) : Submodule.restrictScalars P.Ring (Algebra.Extension.Cotangent.map f).ker = Submodule.map Algebra.Extension.Cotangent.mk (Submodule.comap (Submodule.subtype P.ker) (RingHom.ker f.toRingHom ⊓ P.ker))
• Algebra.Generators.ker_eq_ker_aeval_val 📋 Mathlib.RingTheory.Extension.Generators
  {R : Type u} {S : Type v} {ι : Type w} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Generators R S ι) : P.ker = RingHom.ker (MvPolynomial.aeval P.val)
• Algebra.Generators.ker_ofAlgHom 📋 Mathlib.RingTheory.Extension.Generators
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {I : Type u_8} (f : MvPolynomial I R →ₐ[R] S) (h : Function.Surjective ⇑f) : (Algebra.Generators.ofAlgHom f h).ker = RingHom.ker f.toRingHom
• Algebra.Generators.map_toComp_ker 📋 Mathlib.RingTheory.Extension.Generators
  {R : Type u} {S : Type v} {ι : Type w} [CommRing R] [CommRing S] [Algebra R S] {ι' : Type u_3} {T : Type u_7} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (Q : Algebra.Generators S T ι') (P : Algebra.Generators R S ι) : Ideal.map (Q.toComp P).toAlgHom P.ker = RingHom.ker (Q.ofComp P).toAlgHom
• Algebra.Presentation.span_range_relation_eq_ker_baseChange 📋 Mathlib.RingTheory.Extension.Presentation.Basic
  {R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (T : Type u_1) [CommRing T] [Algebra R T] (P : Algebra.Presentation R S ι σ) : Ideal.span (Set.range fun i => (MvPolynomial.map (algebraMap R T)) (P.relation i)) = RingHom.ker (MvPolynomial.aeval (Algebra.Generators.baseChange T P.toGenerators).val)
• KaehlerDifferential.cotangentComplexBaseChange 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  (R : Type u) (S : Type v) [CommRing R] [CommRing S] (P : Type u_2) (A : Type u_3) [CommRing P] [CommRing A] [Algebra P S] [Algebra P A] [Algebra R P] [Algebra S A] [IsScalarTower P S A] : TensorProduct P A ↥(RingHom.ker (algebraMap P S)) →ₗ[A] TensorProduct P A Ω[P⁄R]
• KaehlerDifferential.cotangentComplexBaseChange_tmul 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] {P : Type u_2} {A : Type u_3} [CommRing P] [CommRing A] [Algebra P S] [Algebra P A] [Algebra R P] [Algebra S A] [IsScalarTower P S A] (a : A) (b : ↥(RingHom.ker (algebraMap P S))) : (KaehlerDifferential.cotangentComplexBaseChange R S P A) (a ⊗ₜ[P] b) = a • (KaehlerDifferential.kerToTensor R P A) ⟨↑b, ⋯⟩
• Algebra.Extension.cotangentComplexBaseChange_eq_lTensor_cotangentComplex 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (A : Type u_1) [CommRing A] [Algebra S A] [Algebra P.Ring A] [IsScalarTower P.Ring S A] : KaehlerDifferential.cotangentComplexBaseChange R S P.Ring A = ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A Ω[P.Ring⁄R]) ∘ₗ LinearMap.baseChange A P.cotangentComplex ∘ₗ ↑((TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A ↥P.ker).symm ≪≫ₗ LinearEquiv.baseChange S A (TensorProduct P.Ring S ↥P.ker) P.Cotangent P.cotangentEquiv)
• Algebra.Extension.lTensor_cotangentComplex_eq_cotangentComplexBaseChange 📋 Mathlib.RingTheory.Extension.Cotangent.Basic
  {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (P : Algebra.Extension R S) (A : Type u_1) [CommRing A] [Algebra S A] [Algebra P.Ring A] [IsScalarTower P.Ring S A] : LinearMap.baseChange A P.cotangentComplex = ↑(TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A Ω[P.Ring⁄R]).symm ∘ₗ KaehlerDifferential.cotangentComplexBaseChange R S P.Ring A ∘ₗ ↑((TensorProduct.AlgebraTensorModule.cancelBaseChange P.Ring S A A ↥P.ker).symm ≪≫ₗ LinearEquiv.baseChange S A (TensorProduct P.Ring S ↥P.ker) P.Cotangent P.cotangentEquiv).symm
• RingHom.ker_fg_of_localizationSpan 📋 Mathlib.RingTheory.Localization.Finiteness
  {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {f : R →+* S} (t : Set R) (ht : Ideal.span t = ⊤) (H : ∀ (g : ↑t), (RingHom.ker (Localization.awayMap f ↑g)).FG) : (RingHom.ker f).FG
• Module.support_of_algebra 📋 Mathlib.RingTheory.Support
  {R : Type u_1} [CommRing R] {A : Type u_3} [Ring A] [Algebra R A] : Module.support R A = PrimeSpectrum.zeroLocus ↑(RingHom.ker (algebraMap R A))
• PrimeSpectrum.preimageEquivFiber_apply_asIdeal 📋 Mathlib.RingTheory.LocalRing.ResidueField.Fiber
  (R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (p : PrimeSpectrum R) (q : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {p})) : ((PrimeSpectrum.preimageEquivFiber R S p) q).asIdeal = RingHom.ker (Algebra.TensorProduct.lift (Ideal.ResidueField.mapₐ p.asIdeal (↑q).asIdeal (Algebra.ofId R S) ⋯) (IsScalarTower.toAlgHom R S (↑q).asIdeal.ResidueField) ⋯).toRingHom
• Module.finite_of_surjective_of_ker_le_nilradical 📋 Mathlib.RingTheory.Finiteness.NilpotentKer
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Module.Finite R T] (f : S →ₐ[R] T) (hf₁ : Function.Surjective ⇑f) (hf₂ : RingHom.ker f ≤ nilradical S) (hf₃ : (RingHom.ker f).FG) : Module.Finite R S
• algebraicIndependent_iff_ker_eq_bot 📋 Mathlib.RingTheory.AlgebraicIndependent.Basic
  {ι : Type u} {R : Type u_2} {A : Type v} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] : AlgebraicIndependent R x ↔ RingHom.ker (MvPolynomial.aeval x).toRingHom = ⊥
• AlgebraicIndependent.repr_ker 📋 Mathlib.RingTheory.AlgebraicIndependent.Basic
  {ι : Type u} {R : Type u_2} {A : Type v} {x : ι → A} [CommRing R] [CommRing A] [Algebra R A] (hx : AlgebraicIndependent R x) : RingHom.ker ↑hx.repr = ⊥
• Algebra.FormallyUnramified.ext' 📋 Mathlib.RingTheory.Unramified.Basic
  {R : Type v} [CommRing R] {A : Type u} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] [Algebra.FormallyUnramified R A] {C : Type u_1} [Ring C] (f : B →+* C) (hf : IsNilpotent (RingHom.ker f)) (g₁ g₂ : A →ₐ[R] B) (h : ∀ (x : A), f (g₁ x) = f (g₂ x)) : g₁ = g₂
• Algebra.FormallyUnramified.lift_unique' 📋 Mathlib.RingTheory.Unramified.Basic
  {R : Type v} [CommRing R] {A : Type u} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] [Algebra.FormallyUnramified R A] {C : Type u_1} [Ring C] [Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent (RingHom.ker ↑f)) (g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59