Jacobson Rings #

The following conditions are equivalent for a ring R:

Every radical ideal I is equal to its Jacobson radical
Every radical ideal I can be written as an intersection of maximal ideals
Every prime ideal I is equal to its Jacobson radical Any ring satisfying any of these equivalent conditions is said to be Jacobson. Some particular examples of Jacobson rings are also proven. isJacobsonRing_quotient says that the quotient of a Jacobson ring is Jacobson. isJacobsonRing_localization says the localization of a Jacobson ring to a single element is Jacobson. isJacobsonRing_polynomial_iff_isJacobsonRing says polynomials over a Jacobson ring form a Jacobson ring.

Main definitions #

Let R be a commutative ring. Jacobson rings are defined using the first of the above conditions

IsJacobsonRing R is the proposition that R is a Jacobson ring. It is a class, implemented as the predicate that for any ideal, I.isRadical implies I.jacobson = I.

Main statements #

isJacobsonRing_iff_prime_eq is the equivalence between conditions 1 and 3 above.
isJacobsonRing_iff_sInf_maximal is the equivalence between conditions 1 and 2 above.
isJacobsonRing_of_surjective says that if R is a Jacobson ring and f : R →+* S is surjective, then S is also a Jacobson ring
MvPolynomial.isJacobsonRing says that multi-variate polynomials over a Jacobson ring are Jacobson.

Tags #

Jacobson, Jacobson Ring

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class IsJacobsonRing (R : Type u_3) [CommRing R] :

Prop

A ring is a Jacobson ring if for every radical ideal I, the Jacobson radical of I is equal to I. See isJacobsonRing_iff_prime_eq and isJacobsonRing_iff_sInf_maximal for equivalent definitions.

out' (I : Ideal R) : I.IsRadical → I.jacobson = I

Instances

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theorem isJacobsonRing_iff {R : Type u_3} [CommRing R] :

IsJacobsonRing R ↔ ∀ (I : Ideal R), I.IsRadical → I.jacobson = I

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theorem IsJacobsonRing.out {R : Type u_3} [CommRing R] :

IsJacobsonRing R → ∀ {I : Ideal R}, I.IsRadical → I.jacobson = I

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theorem isJacobsonRing_iff_prime_eq {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ (P : Ideal R), P.IsPrime → P.jacobson = P

A ring is a Jacobson ring if and only if for all prime ideals P, the Jacobson radical of P is equal to P.

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theorem isJacobsonRing_iff_sInf_maximal {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ (M : Set (Ideal R)), (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M

A ring R is Jacobson if and only if for every prime ideal I, I can be written as the infimum of some collection of maximal ideals. Allowing ⊤ in the set M of maximal ideals is equivalent, but makes some proofs cleaner.

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theorem isJacobsonRing_iff_sInf_maximal' {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ (M : Set (Ideal R)), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M

A variant of isJacobsonRing_iff_sInf_maximal with a different spelling of "maximal or ⊤".

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theorem Ideal.radical_eq_jacobson {R : Type u_1} [CommRing R] [H : IsJacobsonRing R] (I : Ideal R) :

I.radical = I.jacobson

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@[instance 100]

instance instIsJacobsonRingOfIsArtinianRing {R : Type u_1} [CommRing R] [IsArtinianRing R] :

IsJacobsonRing R

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theorem isJacobsonRing_of_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [H : IsJacobsonRing R] :

(∃ (f : R →+* S), Function.Surjective ⇑f) → IsJacobsonRing S

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@[instance 100]

instance isJacobsonRing_quotient {R : Type u_1} [CommRing R] {I : Ideal R} [IsJacobsonRing R] :

IsJacobsonRing (R ⧸ I)

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theorem isJacobsonRing_iso {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (e : R ≃+* S) :

IsJacobsonRing R ↔ IsJacobsonRing S

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theorem isJacobsonRing_of_isIntegral {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] [IsJacobsonRing R] :

IsJacobsonRing S

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theorem isJacobsonRing_of_isIntegral' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) [IsJacobsonRing R] :

IsJacobsonRing S

A variant of isJacobsonRing_of_isIntegral that takes RingHom.IsIntegral instead.

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theorem IsLocalization.isMaximal_iff_isMaximal_disjoint {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : IsJacobsonRing R] (J : Ideal S) :

J.IsMaximal ↔ (Ideal.comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J

If R is a Jacobson ring, then maximal ideals in the localization at y correspond to maximal ideals in the original ring R that don't contain y. This lemma gives the correspondence in the particular case of an ideal and its comap. See le_relIso_of_maximal for the more general relation isomorphism

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theorem IsLocalization.isMaximal_of_isMaximal_disjoint {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [IsJacobsonRing R] (I : Ideal R) (hI : I.IsMaximal) (hy : y ∉ I) :

(Ideal.map (algebraMap R S) I).IsMaximal

If R is a Jacobson ring, then maximal ideals in the localization at y correspond to maximal ideals in the original ring R that don't contain y. This lemma gives the correspondence in the particular case of an ideal and its map. See le_relIso_of_maximal for the more general statement, and the reverse of this implication

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def IsLocalization.orderIsoOfMaximal {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [IsJacobsonRing R] :

{ p : Ideal S // p.IsMaximal } ≃o { p : Ideal R // p.IsMaximal ∧ y ∉ p }

If R is a Jacobson ring, then maximal ideals in the localization at y correspond to maximal ideals in the original ring R that don't contain y

Equations

One or more equations did not get rendered due to their size.

Instances For

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theorem isJacobsonRing_localization {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : IsJacobsonRing R] :

IsJacobsonRing S

If S is the localization of the Jacobson ring R at the submonoid generated by y : R, then S is Jacobson.

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theorem Polynomial.Subring.mem_closure_image_of {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (g : S →+* T) (u : Set S) (x : S) (hx : x ∈ Subring.closure u) :

g x ∈ Subring.closure (⇑g '' u)

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theorem Polynomial.mem_closure_X_union_C {R : Type u_1} [Ring R] (p : Polynomial R) :

p ∈ Subring.closure (insert Polynomial.X {f : Polynomial R | f.degree ≤ 0})

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theorem Polynomial.isIntegral_isLocalization_polynomial_quotient {R : Type u_1} [CommRing R] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [CommRing Sₘ] (P : Ideal (Polynomial R)) (pX : Polynomial R) (hpX : pX ∈ P) [Algebra (R ⧸ Ideal.comap Polynomial.C P) Rₘ] [IsLocalization.Away (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff Rₘ] [Algebra (Polynomial R ⧸ P) Sₘ] [IsLocalization (Submonoid.map (Ideal.quotientMap P Polynomial.C ⋯) (Submonoid.powers (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff)) Sₘ] :

(IsLocalization.map Sₘ (Ideal.quotientMap P Polynomial.C ⋯) ⋯).IsIntegral

If I is a prime ideal of R[X] and pX ∈ I is a non-constant polynomial, then the map R →+* R[x]/I descends to an integral map when localizing at pX.leadingCoeff. In particular X is integral because it satisfies pX, and constants are trivially integral, so integrality of the entire extension follows by closure under addition and multiplication.

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theorem Polynomial.jacobson_bot_of_integral_localization {S : Type u_2} [CommRing S] [IsDomain S] {R : Type u_5} [CommRing R] [IsDomain R] [IsJacobsonRing R] (Rₘ : Type u_6) (Sₘ : Type u_7) [CommRing Rₘ] [CommRing Sₘ] (φ : R →+* S) (hφ : Function.Injective ⇑φ) (x : R) (hx : x ≠ 0) [Algebra R Rₘ] [IsLocalization.Away x Rₘ] [Algebra S Sₘ] [IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ] (hφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral) :

⊥.jacobson = ⊥

If f : R → S descends to an integral map in the localization at x, and R is a Jacobson ring, then the intersection of all maximal ideals in S is trivial

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theorem Polynomial.isJacobsonRing_polynomial_of_isJacobsonRing {R : Type u_1} [CommRing R] (hR : IsJacobsonRing R) :

IsJacobsonRing (Polynomial R)

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theorem Polynomial.isJacobsonRing_polynomial_iff_isJacobsonRing {R : Type u_1} [CommRing R] :

IsJacobsonRing (Polynomial R) ↔ IsJacobsonRing R

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instance Polynomial.instIsJacobsonRing {R : Type u_1} [CommRing R] [IsJacobsonRing R] :

IsJacobsonRing (Polynomial R)

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theorem Polynomial.isMaximal_comap_C_of_isMaximal {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] [Nontrivial R] (hP' : ∀ (x : R), Polynomial.C x ∈ P → x = 0) :

(Ideal.comap Polynomial.C P).IsMaximal

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theorem Polynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] :

((Ideal.Quotient.mk P).comp Polynomial.C).IsIntegral

If R is a Jacobson ring, and P is a maximal ideal of R[X], then R → R[X]/P is an integral map.

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theorem Polynomial.isMaximal_comap_C_of_isJacobsonRing {R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) [hP : P.IsMaximal] [IsJacobsonRing R] :

(Ideal.comap Polynomial.C P).IsMaximal

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theorem Polynomial.comp_C_integral_of_surjective_of_isJacobsonRing {R : Type u_1} [CommRing R] [IsJacobsonRing R] {S : Type u_2} [Field S] (f : Polynomial R →+* S) (hf : Function.Surjective ⇑f) :

(f.comp Polynomial.C).IsIntegral

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theorem MvPolynomial.isJacobsonRing_MvPolynomial_fin {R : Type u} [CommRing R] [H : IsJacobsonRing R] (n : ℕ) :

IsJacobsonRing (MvPolynomial (Fin n) R)

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instance MvPolynomial.isJacobsonRing {R : Type u_1} [CommRing R] {ι : Type u_2} [Finite ι] [IsJacobsonRing R] :

IsJacobsonRing (MvPolynomial ι R)

General form of the Nullstellensatz for Jacobson rings, since in a Jacobson ring we have Inf {P maximal | P ≥ I} = Inf {P prime | P ≥ I} = I.radical. Fields are always Jacobson, and in that special case this is (most of) the classical Nullstellensatz, since I(V(I)) is the intersection of maximal ideals containing I, which is then I.radical

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theorem MvPolynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing {n : ℕ} {R : Type u_1} [CommRing R] [IsJacobsonRing R] (P : Ideal (MvPolynomial (Fin n) R)) [hP : P.IsMaximal] :

((Ideal.Quotient.mk P).comp MvPolynomial.C).IsIntegral

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theorem MvPolynomial.comp_C_integral_of_surjective_of_isJacobsonRing {R : Type u_1} [CommRing R] [IsJacobsonRing R] {σ : Type u_2} [Finite σ] {S : Type u_3} [Field S] (f : MvPolynomial σ R →+* S) (hf : Function.Surjective ⇑f) :

(f.comp MvPolynomial.C).IsIntegral

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theorem isJacobsonRing_of_finiteType {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [IsJacobsonRing A] [Algebra.FiniteType A B] :

IsJacobsonRing B

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theorem RingHom.FiniteType.isJacobsonRing {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] {f : A →+* B} [IsJacobsonRing A] (H : f.FiniteType) :

IsJacobsonRing B

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theorem finite_of_finite_type_of_isJacobsonRing (R : Type u_1) (S : Type u_2) [CommRing R] [Field S] [Algebra R S] [IsJacobsonRing R] [Algebra.FiniteType R S] :

Module.Finite R S

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theorem RingHom.finite_iff_finiteType_of_isJacobsonRing {R : Type u_1} {S : Type u_2} [CommRing R] [IsJacobsonRing R] [Field S] {f : R →+* S} :

f.Finite ↔ f.FiniteType

If f : R →+* S is a ring homomorphism from a jacobson ring to a field, then it is finite if and only if it is finite type.

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@[deprecated IsJacobsonRing]

def Ideal.IsJacobson (R : Type u_3) [CommRing R] :

Prop

Alias of IsJacobsonRing.

Equations

Ideal.IsJacobson = IsJacobsonRing

Instances For

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@[deprecated isJacobsonRing_iff]

theorem Ideal.isJacobson_iff {R : Type u_3} [CommRing R] :

IsJacobsonRing R ↔ ∀ (I : Ideal R), I.IsRadical → I.jacobson = I

Alias of isJacobsonRing_iff.

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@[deprecated IsJacobsonRing.out]

theorem Ideal.IsJacobson.out {R : Type u_3} [CommRing R] :

IsJacobsonRing R → ∀ {I : Ideal R}, I.IsRadical → I.jacobson = I

Alias of IsJacobsonRing.out.

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@[deprecated isJacobsonRing_iff_prime_eq]

theorem Ideal.isJacobson_iff_prime_eq {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ (P : Ideal R), P.IsPrime → P.jacobson = P

Alias of isJacobsonRing_iff_prime_eq.

A ring is a Jacobson ring if and only if for all prime ideals P, the Jacobson radical of P is equal to P.

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@[deprecated isJacobsonRing_iff_sInf_maximal]

theorem Ideal.isJacobson_iff_sInf_maximal {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ (M : Set (Ideal R)), (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M

Alias of isJacobsonRing_iff_sInf_maximal.

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@[deprecated isJacobsonRing_iff_sInf_maximal']

theorem Ideal.isJacobson_iff_sInf_maximal' {R : Type u_1} [CommRing R] :

IsJacobsonRing R ↔ ∀ {I : Ideal R}, I.IsPrime → ∃ (M : Set (Ideal R)), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M

Alias of isJacobsonRing_iff_sInf_maximal'.

A variant of isJacobsonRing_iff_sInf_maximal with a different spelling of "maximal or ⊤".

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@[deprecated isJacobsonRing_of_surjective]

theorem Ideal.isJacobson_of_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [H : IsJacobsonRing R] :

(∃ (f : R →+* S), Function.Surjective ⇑f) → IsJacobsonRing S

Alias of isJacobsonRing_of_surjective.

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@[deprecated isJacobsonRing_iso]

theorem Ideal.isJacobson_iso {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (e : R ≃+* S) :

IsJacobsonRing R ↔ IsJacobsonRing S

Alias of isJacobsonRing_iso.

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@[deprecated isJacobsonRing_of_isIntegral]

theorem Ideal.isJacobson_of_isIntegral {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] [IsJacobsonRing R] :

IsJacobsonRing S

Alias of isJacobsonRing_of_isIntegral.

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@[deprecated isJacobsonRing_of_isIntegral']

theorem Ideal.isJacobson_of_isIntegral' {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) [IsJacobsonRing R] :

IsJacobsonRing S

Alias of isJacobsonRing_of_isIntegral'.

A variant of isJacobsonRing_of_isIntegral that takes RingHom.IsIntegral instead.

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@[deprecated IsLocalization.isMaximal_iff_isMaximal_disjoint]

theorem Ideal.isMaximal_iff_isMaximal_disjoint {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : IsJacobsonRing R] (J : Ideal S) :

J.IsMaximal ↔ (Ideal.comap (algebraMap R S) J).IsMaximal ∧ y ∉ Ideal.comap (algebraMap R S) J

Alias of IsLocalization.isMaximal_iff_isMaximal_disjoint.

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@[deprecated IsLocalization.isMaximal_of_isMaximal_disjoint]

theorem Ideal.isMaximal_of_isMaximal_disjoint {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [IsJacobsonRing R] (I : Ideal R) (hI : I.IsMaximal) (hy : y ∉ I) :

(Ideal.map (algebraMap R S) I).IsMaximal

Alias of IsLocalization.isMaximal_of_isMaximal_disjoint.

If R is a Jacobson ring, then maximal ideals in the localization at y correspond to maximal ideals in the original ring R that don't contain y. This lemma gives the correspondence in the particular case of an ideal and its map. See le_relIso_of_maximal for the more general statement, and the reverse of this implication

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@[deprecated isJacobsonRing_localization]

theorem Ideal.isJacobson_localization {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (y : R) [Algebra R S] [IsLocalization.Away y S] [H : IsJacobsonRing R] :

IsJacobsonRing S

Alias of isJacobsonRing_localization.

If S is the localization of the Jacobson ring R at the submonoid generated by y : R, then S is Jacobson.

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@[deprecated Polynomial.isIntegral_isLocalization_polynomial_quotient]

theorem Ideal.Polynomial.isIntegral_isLocalization_polynomial_quotient {R : Type u_1} [CommRing R] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [CommRing Sₘ] (P : Ideal (Polynomial R)) (pX : Polynomial R) (hpX : pX ∈ P) [Algebra (R ⧸ Ideal.comap Polynomial.C P) Rₘ] [IsLocalization.Away (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff Rₘ] [Algebra (Polynomial R ⧸ P) Sₘ] [IsLocalization (Submonoid.map (Ideal.quotientMap P Polynomial.C ⋯) (Submonoid.powers (Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) pX).leadingCoeff)) Sₘ] :

(IsLocalization.map Sₘ (Ideal.quotientMap P Polynomial.C ⋯) ⋯).IsIntegral

Alias of Polynomial.isIntegral_isLocalization_polynomial_quotient.

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@[deprecated Polynomial.jacobson_bot_of_integral_localization]

theorem Ideal.Polynomial.jacobson_bot_of_integral_localization {S : Type u_2} [CommRing S] [IsDomain S] {R : Type u_5} [CommRing R] [IsDomain R] [IsJacobsonRing R] (Rₘ : Type u_6) (Sₘ : Type u_7) [CommRing Rₘ] [CommRing Sₘ] (φ : R →+* S) (hφ : Function.Injective ⇑φ) (x : R) (hx : x ≠ 0) [Algebra R Rₘ] [IsLocalization.Away x Rₘ] [Algebra S Sₘ] [IsLocalization (Submonoid.map φ (Submonoid.powers x)) Sₘ] (hφ' : (IsLocalization.map Sₘ φ ⋯).IsIntegral) :

⊥.jacobson = ⊥

Alias of Polynomial.jacobson_bot_of_integral_localization.

If f : R → S descends to an integral map in the localization at x, and R is a Jacobson ring, then the intersection of all maximal ideals in S is trivial

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@[deprecated Polynomial.isJacobsonRing_polynomial_of_isJacobsonRing]