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.isMaximal_iff (R : Type u_1) [CommSemiring R] [IsLocalRing R] {I : Ideal R} : I.IsMaximal ↔ I = maximalIdeal R

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.of_singleton_maximalSpectrum {R : Type u_1} [CommSemiring R] [Subsingleton (MaximalSpectrum R)] [Nonempty (MaximalSpectrum R)] : IsLocalRing R

If the maximal spectrum of a ring is a singleton, then the ring is local.

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

theorem IsLocalRing.le_maximalIdeal_of_isPrime {R : Type u_1} [CommSemiring R] [IsLocalRing R] (p : Ideal R) [hp : p.IsPrime] : p ≤ maximalIdeal R

theorem IsLocalRing.notMem_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.

@[deprecated IsLocalRing.notMem_maximalIdeal (since := "2025-05-23")]

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

Alias of IsLocalRing.notMem_maximalIdeal.

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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 = ⊥

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

theorem IsLocalRing.ker_eq_maximalIdeal {R : Type u_1} {K : Type u_3} [CommRing R] [IsLocalRing R] [DivisionRing 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 = ⊥