# Integral closure as a characteristic predicate

We prove basic properties of IsIntegralClosure.

theorem IsIntegral.isUnit {R : Type u_1} {S : Type u_2} [Field R] [Ring S] [IsDomain S] [Algebra R S] {x : S} (int : IsIntegral R x) (h0 : x ≠ 0) : IsUnit x

A nonzero element in a domain integral over a field is a unit.

theorem isField_of_isIntegral_of_isField' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] [Algebra.IsIntegral R S] (hR : IsField R) : IsField S

A commutative domain that is an integral algebra over a field is a field.

theorem IsIntegral.inv_mem_adjoin {R : Type u_1} {S : Type u_2} [Field R] [DivisionRing S] [Algebra R S] {x : S} (int : IsIntegral R x) : x⁻¹ ∈ Algebra.adjoin R {x}

theorem IsIntegral.inv_mem {R : Type u_1} {S : Type u_2} [Field R] [DivisionRing S] [Algebra R S] {x : S} {A : Subalgebra R S} (int : IsIntegral R x) (hx : x ∈ A) : x⁻¹ ∈ A

The inverse of an integral element in a subalgebra of a division ring over a field also lies in that subalgebra.

theorem Algebra.IsIntegral.inv_mem {R : Type u_1} {S : Type u_2} [Field R] [DivisionRing S] [Algebra R S] {x : S} {A : Subalgebra R S} [Algebra.IsIntegral R ↥A] (hx : x ∈ A) : x⁻¹ ∈ A

An integral subalgebra of a division ring over a field is closed under inverses.

theorem IsIntegral.inv {R : Type u_1} {S : Type u_2} [Field R] [DivisionRing S] [Algebra R S] {x : S} (int : IsIntegral R x) : IsIntegral R x⁻¹

The inverse of an integral element in a division ring over a field is also integral.

theorem IsIntegral.mem_of_inv_mem {R : Type u_1} {S : Type u_2} [Field R] [DivisionRing S] [Algebra R S] {x : S} {A : Subalgebra R S} (int : IsIntegral R x) (inv_mem : x⁻¹ ∈ A) : x ∈ A

theorem Algebra.IsIntegral.finite {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [h' : Algebra.FiniteType R A] : Module.Finite R A

The Kurosh problem asks to show that this is still true when A is not necessarily commutative and R is a field, but it has been solved in the negative. See https://arxiv.org/pdf/1706.02383.pdf for criteria for a finitely generated algebraic (= integral) algebra over a field to be finite dimensional.

This could be an instance, but we tend to go from Module.Finite to IsIntegral/IsAlgebraic, and making it an instance will cause the search to be complicated a lot.

theorem Algebra.finite_iff_isIntegral_and_finiteType {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A

finite = integral + finite type

theorem RingHom.IsIntegral.to_finite {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite

theorem RingHom.Finite.of_isIntegral_of_finiteType {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite

Alias of RingHom.IsIntegral.to_finite.

theorem RingHom.finite_iff_isIntegral_and_finiteType {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} : f.Finite ↔ f.IsIntegral ∧ f.FiniteType

finite = integral + finite type

theorem mem_integralClosure_iff_mem_fg (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] {r : A} : r ∈ integralClosure R A ↔ ∃ (M : Subalgebra R A), (Subalgebra.toSubmodule M).FG ∧ r ∈ M

theorem adjoin_le_integralClosure {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} (hx : IsIntegral R x) : Algebra.adjoin R {x} ≤ integralClosure R A

theorem le_integralClosure_iff_isIntegral {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S : Subalgebra R A} : S ≤ integralClosure R A ↔ Algebra.IsIntegral R ↥S

theorem Algebra.IsIntegral.adjoin {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S : Set A} (hS : ∀ x ∈ S, IsIntegral R x) : Algebra.IsIntegral R ↥(Algebra.adjoin R S)

theorem integralClosure_eq_top_iff {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : integralClosure R A = ⊤ ↔ Algebra.IsIntegral R A

theorem Algebra.isIntegral_sup {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S T : Subalgebra R A} : Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T

theorem Algebra.isIntegral_iSup {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {ι : Sort u_5} (S : ι → Subalgebra R A) : Algebra.IsIntegral R ↥(iSup S) ↔ ∀ (i : ι), Algebra.IsIntegral R ↥(S i)

theorem integralClosure_map_algEquiv {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) : Subalgebra.map (↑f) (integralClosure R A) = integralClosure R S

Mapping an integral closure along an AlgEquiv gives the integral closure.

def AlgHom.mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A →ₐ[R] S) : ↥(integralClosure R A) →ₐ[R] ↥(integralClosure R S)

An AlgHom between two rings restrict to an AlgHom between the integral closures inside them.

Equations

• f.mapIntegralClosure = (AlgHom.restrictDomain (↥(integralClosure R A)) f).codRestrict (integralClosure R S) ⋯

@[simp]

theorem AlgHom.coe_mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A →ₐ[R] S) (x : ↥(integralClosure R A)) : ↑(f.mapIntegralClosure x) = f ↑x

def AlgEquiv.mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) : ↥(integralClosure R A) ≃ₐ[R] ↥(integralClosure R S)

An AlgEquiv between two rings restrict to an AlgEquiv between the integral closures inside them.

Equations

• f.mapIntegralClosure = AlgEquiv.ofAlgHom (↑f).mapIntegralClosure (↑f.symm).mapIntegralClosure ⋯ ⋯

@[simp]

theorem AlgEquiv.coe_mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) (x : ↥(integralClosure R A)) : ↑(f.mapIntegralClosure x) = f ↑x

theorem integralClosure.isIntegral {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : ↥(integralClosure R A)) : IsIntegral R x

instance integralClosure.AlgebraIsIntegral {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : Algebra.IsIntegral R ↥(integralClosure R A)

Equations

• ⋯ = ⋯

theorem IsIntegral.of_mul_unit {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x y : B} {r : R} (hr : (algebraMap R B) r * y = 1) (hx : IsIntegral R (x * y)) : IsIntegral R x

theorem RingHom.IsIntegralElem.of_mul_unit {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (x y : S) (r : R) (hr : f r * y = 1) (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x

theorem IsIntegral.of_mem_closure' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) (x : A) : x ∈ Subring.closure G → IsIntegral R x

Generalization of IsIntegral.of_mem_closure bootstrapped up from that lemma

theorem IsIntegral.of_mem_closure'' {R : Type u_1} [CommRing R] {S : Type u_5} [CommRing S] {f : R →+* S} (G : Set S) (hG : ∀ x ∈ G, f.IsIntegralElem x) (x : S) : x ∈ Subring.closure G → f.IsIntegralElem x

theorem IsIntegral.pow {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n)

theorem IsIntegral.nsmul {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x)

theorem IsIntegral.zsmul {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x)

theorem IsIntegral.multiset_prod {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) : IsIntegral R s.prod

theorem IsIntegral.multiset_sum {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) : IsIntegral R s.sum

theorem IsIntegral.prod {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {α : Type u_5} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) : IsIntegral R (∏ x ∈ s, f x)

theorem IsIntegral.sum {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {α : Type u_5} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) : IsIntegral R (∑ x ∈ s, f x)

theorem IsIntegral.det {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {n : Type u_5} [Fintype n] [DecidableEq n] {M : Matrix n n A} (h : ∀ (i j : n), IsIntegral R (M i j)) : IsIntegral R M.det

@[simp]

theorem IsIntegral.pow_iff {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x

theorem IsIntegral.tmul {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y)

noncomputable def normalizeScaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) : Polynomial R

The monic polynomial whose roots are p.leadingCoeff * x for roots x of p.

Equations

• normalizeScaleRoots p = ∑ i ∈ p.support, (Polynomial.monomial i) (if i = p.natDegree then 1 else p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i))

theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow {R : Type u_1} [CommRing R] (p : Polynomial R) (i : ℕ) (hp : 1 ≤ p.natDegree) : (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1)

theorem leadingCoeff_smul_normalizeScaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) : p.leadingCoeff • normalizeScaleRoots p = p.scaleRoots p.leadingCoeff

theorem normalizeScaleRoots_support {R : Type u_1} [CommRing R] (p : Polynomial R) : (normalizeScaleRoots p).support ≤ p.support

theorem normalizeScaleRoots_degree {R : Type u_1} [CommRing R] (p : Polynomial R) : (normalizeScaleRoots p).degree = p.degree

theorem normalizeScaleRoots_eval₂_leadingCoeff_mul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (p : Polynomial R) (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) : Polynomial.eval₂ f (f p.leadingCoeff * x) (normalizeScaleRoots p) = f p.leadingCoeff ^ (p.natDegree - 1) * Polynomial.eval₂ f x p

theorem normalizeScaleRoots_monic {R : Type u_1} [CommRing R] (p : Polynomial R) (h : p ≠ 0) : (normalizeScaleRoots p).Monic

theorem RingHom.isIntegralElem_leadingCoeff_mul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (p : Polynomial R) (x : S) (h : Polynomial.eval₂ f x p = 0) : f.IsIntegralElem (f p.leadingCoeff * x)

Given a p : R[X] and a x : S such that p.eval₂ f x = 0, f p.leadingCoeff * x is integral.

theorem isIntegral_leadingCoeff_smul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (p : Polynomial R) (x : S) [Algebra R S] (h : (Polynomial.aeval x) p = 0) : IsIntegral R (p.leadingCoeff • x)

Given a p : R[X] and a root x : S, then p.leadingCoeff • x : S is integral over R.

theorem Polynomial.Monic.quotient_isIntegralElem {S : Type u_4} [CommRing S] {g : Polynomial S} (mon : g.Monic) {I : Ideal (Polynomial S)} (h : g ∈ I) : ((Ideal.Quotient.mk I).comp (algebraMap S (Polynomial S))).IsIntegralElem ((Ideal.Quotient.mk I) Polynomial.X)

theorem Polynomial.Monic.quotient_isIntegral {S : Type u_4} [CommRing S] {g : Polynomial S} (mon : g.Monic) {I : Ideal (Polynomial S)} (h : g ∈ I) : ((Ideal.Quotient.mkₐ S I).comp (Algebra.ofId S (Polynomial S))).IsIntegral

instance integralClosure.isIntegralClosure (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] : IsIntegralClosure (↥(integralClosure R A)) R A

Equations

• ⋯ = ⋯

theorem IsIntegralClosure.algebraMap_injective (A : Type u_1) (R : Type u_2) (B : Type u_3) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] : Function.Injective ⇑(algebraMap A B)

theorem IsIntegralClosure.isIntegral (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x

theorem IsIntegralClosure.isIntegral_algebra (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] : Algebra.IsIntegral R A

theorem IsIntegralClosure.noZeroSMulDivisors (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] : NoZeroSMulDivisors R A

noncomputable def IsIntegralClosure.mk' {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (hx : IsIntegral R x) : A

If x : B is integral over R, then it is an element of the integral closure of R in B.

Equations

• IsIntegralClosure.mk' A x hx = Classical.choose ⋯

@[simp]

theorem IsIntegralClosure.algebraMap_mk' {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (hx : IsIntegral R x) : (algebraMap A B) (IsIntegralClosure.mk' A x hx) = x

@[simp]

theorem IsIntegralClosure.mk'_one {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (h : IsIntegral R 1 := ⋯) : IsIntegralClosure.mk' A 1 h = 1

@[simp]

theorem IsIntegralClosure.mk'_zero {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (h : IsIntegral R 0 := ⋯) : IsIntegralClosure.mk' A 0 h = 0

theorem IsIntegralClosure.mk'_add {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegralClosure.mk' A (x + y) ⋯ = IsIntegralClosure.mk' A x hx + IsIntegralClosure.mk' A y hy

theorem IsIntegralClosure.mk'_mul {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegralClosure.mk' A (x * y) ⋯ = IsIntegralClosure.mk' A x hx * IsIntegralClosure.mk' A y hy

@[simp]

theorem IsIntegralClosure.mk'_algebraMap {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] (x : R) (h : IsIntegral R ((algebraMap R B) x) := ⋯) : IsIntegralClosure.mk' A ((algebraMap R B) x) h = (algebraMap R A) x

theorem IsIntegralClosure.isField {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] [IsDomain A] (hR : IsField R) : IsField A

The integral closure of a field in a commutative domain is always a field.

noncomputable def IsIntegralClosure.lift (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] {S : Type u_4} [CommRing S] [Algebra R S] [Algebra S B] [IsScalarTower R S B] [Algebra R A] [IsScalarTower R A B] [isIntegral : Algebra.IsIntegral R S] : S →ₐ[R] A

If B / S / R is a tower of ring extensions where S is integral over R, then S maps (uniquely) into an integral closure B / A / R.

Equations

• IsIntegralClosure.lift R A B = { toFun := fun (x : S) => IsIntegralClosure.mk' A ((algebraMap S B) x) ⋯, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem IsIntegralClosure.algebraMap_lift (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] {S : Type u_4} [CommRing S] [Algebra R S] [Algebra S B] [IsScalarTower R S B] [Algebra R A] [IsScalarTower R A B] [isIntegral : Algebra.IsIntegral R S] (x : S) : (algebraMap A B) ((IsIntegralClosure.lift R A B) x) = (algebraMap S B) x

noncomputable def IsIntegralClosure.equiv (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (A' : Type u_4) [CommRing A'] [Algebra A' B] [IsIntegralClosure A' R B] [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B] : A ≃ₐ[R] A'

Integral closures are all isomorphic to each other.

Equations

• IsIntegralClosure.equiv R A B A' = AlgEquiv.ofAlgHom (IsIntegralClosure.lift R A' B) (IsIntegralClosure.lift R A B) ⋯ ⋯

@[simp]

theorem IsIntegralClosure.algebraMap_equiv (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (A' : Type u_4) [CommRing A'] [Algebra A' B] [IsIntegralClosure A' R B] [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B] (x : A) : (algebraMap A' B) ((IsIntegralClosure.equiv R A B A') x) = (algebraMap A B) x

theorem isIntegral_trans {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] [Algebra.IsIntegral R A] (x : B) (hx : IsIntegral A x) : IsIntegral R x

If A is an R-algebra all of whose elements are integral over R, and x is an element of an A-algebra that is integral over A, then x is integral over R.

theorem Algebra.IsIntegral.trans {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] [Algebra.IsIntegral R A] [Algebra.IsIntegral A B] : Algebra.IsIntegral R B

If A is an R-algebra all of whose elements are integral over R, and B is an A-algebra all of whose elements are integral over A, then all elements of B are integral over R.

theorem RingHom.IsIntegral.trans {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral

theorem IsIntegralClosure.tower_top {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {B : Type u_6} {C : Type u_7} [CommRing C] [CommRing B] [Algebra R B] [Algebra A B] [Algebra C B] [IsScalarTower R A B] [IsIntegralClosure C R B] [Algebra.IsIntegral R A] : IsIntegralClosure C A B

If R → A → B is an algebra tower, C is the integral closure of R in B and A is integral over R, then C is the integral closure of A in B.

theorem RingHom.isIntegral_of_surjective {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) : f.IsIntegral

theorem Algebra.isIntegral_of_surjective {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (h : Function.Surjective ⇑(algebraMap R A)) : Algebra.IsIntegral R A

theorem IsIntegral.tower_bot {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] (H : Function.Injective ⇑(algebraMap A B)) {x : A} (h : IsIntegral R ((algebraMap A B) x)) : IsIntegral R x

If R → A → B is an algebra tower with A → B injective, then if the entire tower is an integral extension so is R → A

theorem RingHom.IsIntegral.tower_bot {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (hg : Function.Injective ⇑g) (hfg : (g.comp f).IsIntegral) : f.IsIntegral

theorem Algebra.IsIntegral.tower_bot {R : Type u_1} {S : Type u_4} (T : Type u_5) [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [NoZeroSMulDivisors S T] [Nontrivial T] [IsScalarTower R S T] [h : Algebra.IsIntegral R T] : Algebra.IsIntegral R S

Let T / S / R be a tower of algebras, T is non-trivial and is a torsion free S-module, then if T is an integral R-algebra, then S is an integral R-algebra.

theorem IsIntegral.tower_bot_of_field {R : Type u_6} {A : Type u_7} {B : Type u_8} [CommRing R] [Field A] [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : A} (h : IsIntegral R ((algebraMap A B) x)) : IsIntegral R x

theorem RingHom.isIntegralElem.of_comp {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) {x : T} (h : (g.comp f).IsIntegralElem x) : g.IsIntegralElem x

theorem RingHom.IsIntegral.tower_top {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (h : (g.comp f).IsIntegral) : g.IsIntegral

theorem Algebra.IsIntegral.tower_top (R : Type u_1) {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [h : Algebra.IsIntegral R T] : Algebra.IsIntegral S T

Let T / S / R be a tower of algebras, T is an integral R-algebra, then it is integral as an S-algebra.

theorem RingHom.IsIntegral.quotient {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} (hf : f.IsIntegral) : (Ideal.quotientMap I f ⋯).IsIntegral

instance instIsIntegralQuotientIdeal {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {I : Ideal A} [Algebra.IsIntegral R A] : Algebra.IsIntegral R (A ⧸ I)

Equations

• ⋯ = ⋯

instance Algebra.IsIntegral.quotient {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {I : Ideal A} [Algebra.IsIntegral R A] : Algebra.IsIntegral (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I)

Equations

• ⋯ = ⋯

theorem isIntegral_quotientMap_iff {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} : (Ideal.quotientMap I f ⋯).IsIntegral ↔ ((Ideal.Quotient.mk I).comp f).IsIntegral

theorem isField_of_isIntegral_of_isField {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (hRS : Function.Injective ⇑(algebraMap R S)) (hS : IsField S) : IsField R

If the integral extension R → S is injective, and S is a field, then R is also a field.

theorem Algebra.IsIntegral.isField_iff_isField {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] [Algebra.IsIntegral R S] (hRS : Function.Injective ⇑(algebraMap R S)) : IsField R ↔ IsField S

theorem integralClosure_idem {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : integralClosure (↥(integralClosure R A)) A = ⊥

instance instIsDomainSubtypeMemSubalgebraIntegralClosure {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] : IsDomain ↥(integralClosure R S)

Equations

• ⋯ = ⋯

theorem roots_mem_integralClosure {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] {f : Polynomial R} (hf : f.Monic) {a : S} (ha : a ∈ f.aroots S) : a ∈ integralClosure R S