Maximal ideal of local rings

We prove basic properties of the maximal ideal of a local ring.

@[simp]

theorem IsLocalRing.mem_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] (x : R) : x ∈ maximalIdeal R ↔ x ∈ nonunits R

instance IsLocalRing.maximalIdeal.isMaximal (R : Type u_1) [CommSemiring R] [IsLocalRing R] : (maximalIdeal R).IsMaximal

theorem IsLocalRing.maximal_ideal_unique (R : Type u_1) [CommSemiring R] [IsLocalRing R] : ∃! I : Ideal R, I.IsMaximal

theorem IsLocalRing.eq_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {I : Ideal R} (hI : I.IsMaximal) : I = maximalIdeal R

instance IsLocalRing.instUniqueMaximalSpectrum {R : Type u_1} [CommSemiring R] [IsLocalRing R] : Unique (MaximalSpectrum R)

The maximal spectrum of a local ring is a singleton.

Equations

• IsLocalRing.instUniqueMaximalSpectrum = { default := { asIdeal := IsLocalRing.maximalIdeal R, isMaximal := ⋯ }, uniq := ⋯ }

theorem IsLocalRing.le_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {J : Ideal R} (hJ : J ≠ ⊤) : J ≤ maximalIdeal R

theorem IsLocalRing.not_mem_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {x : R} : x ∉ maximalIdeal R ↔ IsUnit x

An element x of a commutative local semiring is not contained in the maximal ideal iff it is a unit.

theorem IsLocalRing.isField_iff_maximalIdeal_eq {R : Type u_1} [CommSemiring R] [IsLocalRing R] : IsField R ↔ maximalIdeal R = ⊥

@[deprecated IsLocalRing.maximal_ideal_unique (since := "2024-11-11")]

theorem LocalRing.maximal_ideal_unique (R : Type u_1) [CommSemiring R] [IsLocalRing R] : ∃! I : Ideal R, I.IsMaximal

Alias of IsLocalRing.maximal_ideal_unique.

@[deprecated IsLocalRing.eq_maximalIdeal (since := "2024-11-11")]

theorem LocalRing.eq_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {I : Ideal R} (hI : I.IsMaximal) : I = IsLocalRing.maximalIdeal R

Alias of IsLocalRing.eq_maximalIdeal.

@[deprecated IsLocalRing.le_maximalIdeal (since := "2024-11-11")]

theorem LocalRing.le_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {J : Ideal R} (hJ : J ≠ ⊤) : J ≤ IsLocalRing.maximalIdeal R

Alias of IsLocalRing.le_maximalIdeal.

@[deprecated IsLocalRing.mem_maximalIdeal (since := "2024-11-11")]

theorem LocalRing.mem_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] (x : R) : x ∈ IsLocalRing.maximalIdeal R ↔ x ∈ nonunits R

Alias of IsLocalRing.mem_maximalIdeal.

@[deprecated IsLocalRing.not_mem_maximalIdeal (since := "2024-11-11")]

theorem LocalRing.not_mem_maximalIdeal {R : Type u_1} [CommSemiring R] [IsLocalRing R] {x : R} : x ∉ IsLocalRing.maximalIdeal R ↔ IsUnit x

Alias of IsLocalRing.not_mem_maximalIdeal.

---

An element x of a commutative local semiring is not contained in the maximal ideal iff it is a unit.

@[deprecated IsLocalRing.isField_iff_maximalIdeal_eq (since := "2024-11-11")]

theorem LocalRing.isField_iff_maximalIdeal_eq {R : Type u_1} [CommSemiring R] [IsLocalRing R] : IsField R ↔ IsLocalRing.maximalIdeal R = ⊥

Alias of IsLocalRing.isField_iff_maximalIdeal_eq.

theorem IsLocalRing.maximalIdeal_le_jacobson {R : Type u_1} [CommRing R] [IsLocalRing R] (I : Ideal R) : maximalIdeal R ≤ I.jacobson

theorem IsLocalRing.jacobson_eq_maximalIdeal {R : Type u_1} [CommRing R] [IsLocalRing R] (I : Ideal R) (h : I ≠ ⊤) : I.jacobson = maximalIdeal R

@[deprecated IsLocalRing.maximalIdeal_le_jacobson (since := "2024-11-11")]

theorem LocalRing.maximalIdeal_le_jacobson {R : Type u_1} [CommRing R] [IsLocalRing R] (I : Ideal R) : IsLocalRing.maximalIdeal R ≤ I.jacobson

Alias of IsLocalRing.maximalIdeal_le_jacobson.

@[deprecated IsLocalRing.jacobson_eq_maximalIdeal (since := "2024-11-11")]

theorem LocalRing.jacobson_eq_maximalIdeal {R : Type u_1} [CommRing R] [IsLocalRing R] (I : Ideal R) (h : I ≠ ⊤) : I.jacobson = IsLocalRing.maximalIdeal R

Alias of IsLocalRing.jacobson_eq_maximalIdeal.

theorem IsLocalRing.ker_eq_maximalIdeal {R : Type u_1} {K : Type u_3} [CommRing R] [IsLocalRing R] [Field K] (φ : R →+* K) (hφ : Function.Surjective ⇑φ) : RingHom.ker φ = maximalIdeal R

theorem IsLocalRing.maximalIdeal_eq_bot {R : Type u_4} [Field R] : maximalIdeal R = ⊥

@[deprecated IsLocalRing.ker_eq_maximalIdeal (since := "2024-11-09")]

theorem LocalRing.ker_eq_maximalIdeal {R : Type u_1} {K : Type u_3} [CommRing R] [IsLocalRing R] [Field K] (φ : R →+* K) (hφ : Function.Surjective ⇑φ) : RingHom.ker φ = IsLocalRing.maximalIdeal R

Alias of IsLocalRing.ker_eq_maximalIdeal.

@[deprecated IsLocalRing.maximalIdeal_eq_bot (since := "2024-11-09")]

theorem LocalRing.maximalIdeal_eq_bot {R : Type u_4} [Field R] : IsLocalRing.maximalIdeal R = ⊥

Alias of IsLocalRing.maximalIdeal_eq_bot.

theorem IsLocalRing.of_nilradical_isMaximal {R : Type u_4} [CommSemiring R] [h : (nilradical R).IsMaximal] : IsLocalRing R

@[deprecated IsLocalRing.of_nilradical_isMaximal (since := "2024-11-09")]

theorem LocalRing.of_nilradical_isMaximal {R : Type u_4} [CommSemiring R] [h : (nilradical R).IsMaximal] : IsLocalRing R

Alias of IsLocalRing.of_nilradical_isMaximal.

noncomputable def localizationEquivSelfOfNilradicalIsMaximal {R : Type u_4} [CommSemiring R] {S : Type u_5} [CommSemiring S] [Algebra R S] {M : Submonoid R} [h : (nilradical R).IsMaximal] (h' : 0 ∉ M) [IsLocalization M S] : R ≃ₐ[R] S

Let S be the localization of a commutative semiring R at a submonoid M that does not contain 0. If the nilradical of R is maximal then there is a R-algebra isomorphism between R and S.

Equations

• localizationEquivSelfOfNilradicalIsMaximal h' = IsLocalization.atUnits R M ⋯