Jacobson radical

The Jacobson radical of a ring R is defined to be the intersection of all maximal ideals of R. This is similar to how the nilradical is equal to the intersection of all prime ideals of R.

We can extend the idea of the nilradical of R to ideals of R, by letting the nilradical of an ideal I be the intersection of prime ideals containing I. Under this extension, the original nilradical is the radical of the zero ideal ⊥. Here we define the Jacobson radical of an ideal I in a similar way, as the intersection of maximal ideals containing I.

Main definitions

Let R be a ring, and I be a left ideal of R

• Ideal.jacobson I is the Jacobson radical, i.e. the infimum of all maximal ideals containing I.

• Ideal.IsLocal I is the proposition that the jacobson radical of I is itself a maximal ideal

Furthermore when I is a two-sided ideal of R

• TwoSidedIdeal.jacobson I is the Jacobson radical as a two-sided ideal

Main statements

• mem_jacobson_iff gives a characterization of members of the jacobson of I

• Ideal.isLocal_of_isMaximal_radical: if the radical of I is maximal then so is the jacobson radical

Tags

Jacobson, Jacobson radical, Local Ideal

def Ideal.jacobson {R : Type u} [Ring R] (I : Ideal R) : Ideal R

The Jacobson radical of I is the infimum of all maximal (left) ideals containing I.

Equations

• I.jacobson = sInf {J : Ideal R | I ≤ J ∧ J.IsMaximal}

theorem Ideal.le_jacobson {R : Type u} [Ring R] {I : Ideal R} : I ≤ I.jacobson

@[simp]

theorem Ideal.jacobson_idem {R : Type u} [Ring R] {I : Ideal R} : I.jacobson.jacobson = I.jacobson

@[simp]

theorem Ideal.jacobson_top {R : Type u} [Ring R] : ⊤.jacobson = ⊤

@[simp]

theorem Ideal.jacobson_eq_top_iff {R : Type u} [Ring R] {I : Ideal R} : I.jacobson = ⊤ ↔ I = ⊤

theorem Ideal.jacobson_eq_bot {R : Type u} [Ring R] {I : Ideal R} : I.jacobson = ⊥ → I = ⊥

theorem Ideal.jacobson_eq_self_of_isMaximal {R : Type u} [Ring R] {I : Ideal R} [H : I.IsMaximal] : I.jacobson = I

@[instance 100]

instance Ideal.jacobson.isMaximal {R : Type u} [Ring R] {I : Ideal R} [H : I.IsMaximal] : I.jacobson.IsMaximal

Equations

• ⋯ = ⋯

theorem Ideal.mem_jacobson_iff {R : Type u} [Ring R] {I : Ideal R} {x : R} : x ∈ I.jacobson ↔ ∀ (y : R), ∃ (z : R), z * y * x + z - 1 ∈ I

theorem Ideal.exists_mul_add_sub_mem_of_mem_jacobson {R : Type u} [Ring R] {I : Ideal R} (r : R) (h : r ∈ I.jacobson) : ∃ (s : R), s * (r + 1) - 1 ∈ I

theorem Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson {R : Type u} [Ring R] {I : Ideal R} (r : R) (h : r - 1 ∈ I.jacobson) : ∃ (s : R), s * r - 1 ∈ I

theorem Ideal.eq_jacobson_iff_sInf_maximal {R : Type u} [Ring R] {I : Ideal R} : I.jacobson = I ↔ ∃ (M : Set (Ideal R)), (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M

An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals. Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs.

theorem Ideal.eq_jacobson_iff_sInf_maximal' {R : Type u} [Ring R] {I : Ideal R} : I.jacobson = I ↔ ∃ (M : Set (Ideal R)), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M

theorem Ideal.eq_jacobson_iff_not_mem {R : Type u} [Ring R] {I : Ideal R} : I.jacobson = I ↔ ∀ x ∉ I, ∃ (M : Ideal R), (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M

An ideal I equals its Jacobson radical if and only if every element outside I also lies outside of a maximal ideal containing I.

theorem Ideal.map_jacobson_of_surjective {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

theorem Ideal.map_jacobson_of_bijective {R : Type u} {S : Type v} [Ring R] [Ring S] {I : Ideal R} {f : R →+* S} (hf : Function.Bijective ⇑f) : Ideal.map f I.jacobson = (Ideal.map f I).jacobson

theorem Ideal.comap_jacobson {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {K : Ideal S} : Ideal.comap f K.jacobson = sInf (Ideal.comap f '' {J : Ideal S | K ≤ J ∧ J.IsMaximal})

theorem Ideal.comap_jacobson_of_surjective {R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} (hf : Function.Surjective ⇑f) {K : Ideal S} : Ideal.comap f K.jacobson = (Ideal.comap f K).jacobson

theorem Ideal.jacobson_mono {R : Type u} [Ring R] {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson

theorem Ideal.jacobson_mul_mem_right {R : Type u} [Ring R] {I : Ideal R} (mul_mem_right : ∀ {x y : R}, x ∈ I → x * y ∈ I) {x y : R} : x ∈ I.jacobson → x * y ∈ I.jacobson

The Jacobson radical of a two-sided ideal is two-sided.

It is preferable to use TwoSidedIdeal.jacobson instead of this lemma.

theorem Ideal.radical_le_jacobson {R : Type u} [CommRing R] {I : Ideal R} : I.radical ≤ I.jacobson

theorem Ideal.isRadical_of_eq_jacobson {R : Type u} [CommRing R] {I : Ideal R} (h : I.jacobson = I) : I.IsRadical

theorem Ideal.isRadical_jacobson {R : Type u} [CommRing R] (I : Ideal R) : I.jacobson.IsRadical

theorem Ideal.isUnit_of_sub_one_mem_jacobson_bot {R : Type u} [CommRing R] (r : R) (h : r - 1 ∈ ⊥.jacobson) : IsUnit r

theorem Ideal.mem_jacobson_bot {R : Type u} [CommRing R] {x : R} : x ∈ ⊥.jacobson ↔ ∀ (y : R), IsUnit (x * y + 1)

theorem Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot {R : Type u} [CommRing R] {I : Ideal R} : I.jacobson = I ↔ ⊥.jacobson = ⊥

An ideal I of R is equal to its Jacobson radical if and only if the Jacobson radical of the quotient ring R/I is the zero ideal

theorem Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot {R : Type u} [CommRing R] {I : Ideal R} : I.radical = I.jacobson ↔ ⊥.radical = ⊥.jacobson

The standard radical and Jacobson radical of an ideal I of R are equal if and only if the nilradical and Jacobson radical of the quotient ring R/I coincide

theorem Ideal.jacobson_radical_eq_jacobson {R : Type u} [CommRing R] {I : Ideal R} : I.radical.jacobson = I.jacobson

theorem Ideal.jacobson_bot_polynomial_le_sInf_map_maximal {R : Type u} [CommRing R] : ⊥.jacobson ≤ sInf (Ideal.map Polynomial.C '' {J : Ideal R | J.IsMaximal})

theorem Ideal.jacobson_bot_polynomial_of_jacobson_bot {R : Type u} [CommRing R] (h : ⊥.jacobson = ⊥) : ⊥.jacobson = ⊥

class Ideal.IsLocal {R : Type u} [CommRing R] (I : Ideal R) : Prop

An ideal I is local iff its Jacobson radical is maximal.

• out : I.jacobson.IsMaximal

  A ring R is local if and only if its jacobson radical is maximal

theorem Ideal.isLocal_iff {R : Type u} [CommRing R] {I : Ideal R} : I.IsLocal ↔ I.jacobson.IsMaximal

theorem Ideal.isLocal_of_isMaximal_radical {R : Type u} [CommRing R] {I : Ideal R} (hi : I.radical.IsMaximal) : I.IsLocal

theorem Ideal.IsLocal.le_jacobson {R : Type u} [CommRing R] {I J : Ideal R} (hi : I.IsLocal) (hij : I ≤ J) (hj : J ≠ ⊤) : J ≤ I.jacobson

theorem Ideal.IsLocal.mem_jacobson_or_exists_inv {R : Type u} [CommRing R] {I : Ideal R} (hi : I.IsLocal) (x : R) : x ∈ I.jacobson ∨ ∃ (y : R), y * x - 1 ∈ I

theorem Ideal.isPrimary_of_isMaximal_radical {R : Type u} [CommRing R] {I : Ideal R} (hi : I.radical.IsMaximal) : I.IsPrimary

def TwoSidedIdeal.jacobson {R : Type u} [Ring R] (I : TwoSidedIdeal R) : TwoSidedIdeal R

The Jacobson radical of I is the infimum of all maximal (left) ideals containing I.

Equations

• I.jacobson = (TwoSidedIdeal.asIdeal I).jacobson.toTwoSided ⋯

theorem TwoSidedIdeal.asIdeal_jacobson {R : Type u} [Ring R] (I : TwoSidedIdeal R) : TwoSidedIdeal.asIdeal I.jacobson = (TwoSidedIdeal.asIdeal I).jacobson

theorem TwoSidedIdeal.mem_jacobson_iff {R : Type u} [Ring R] {x : R} {I : TwoSidedIdeal R} : x ∈ I.jacobson ↔ ∀ (y : R), ∃ (z : R), z * y * x + z - 1 ∈ I