Documentation

Mathlib.RingTheory.Ideal.Pointwise

Pointwise instances on Ideals #

This file provides the action Ideal.pointwiseMulAction which morally matches the action of mulActionSet (though here an extra Ideal.span is inserted).

This actions is available in the Pointwise locale.

Implementation notes #

This file is similar (but not identical) to Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean. Where possible, try to keep them in sync.

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def Ideal.pointwiseDistribMulAction {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] :
DistribMulAction M (Ideal R)

The action on an ideal corresponding to applying the action to every element.

This is available as an instance in the Pointwise locale.

Equations
Instances For
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    def Ideal.pointwiseMulSemiringAction {M : Type u_1} [Monoid M] {R : Type u_3} [CommRing R] [MulSemiringAction M R] :
    MulSemiringAction M (Ideal R)

    The action on an ideal corresponding to applying the action to every element.

    This is available as an instance in the Pointwise locale.

    Equations
    Instances For
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      theorem Ideal.pointwise_smul_def {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] {a : M} (S : Ideal R) :
      a • S = Ideal.map (MulSemiringAction.toRingHom M R a) S
      source
      theorem Ideal.smul_mem_pointwise_smul {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (m : M) (r : R) (S : Ideal R) :
      r ∈ S → m • r ∈ m • S
      source
      instance Ideal.instCovariantClassHSMulLe {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] :
      CovariantClass M (Ideal R) HSMul.hSMul LE.le
      source
      @[simp]
      theorem Ideal.smul_bot {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) :
      a • ⊥ = ⊥
      source
      theorem Ideal.smul_sup {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (S T : Ideal R) :
      a • (S ⊔ T) = a • S ⊔ a • T
      source
      theorem Ideal.smul_closure {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (s : Set R) :
      a • Ideal.span s = Ideal.span (a • s)
      source
      instance Ideal.pointwise_central_scalar {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] [MulSemiringAction Mᵐᵒᵖ R] [IsCentralScalar M R] :
      IsCentralScalar M (Ideal R)
      source
      @[simp]
      theorem Ideal.pointwise_smul_toAddSubmonoid {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (S : Ideal R) (ha : Function.Surjective fun (r : R) => a • r) :
      (a • S).toAddSubmonoid = a • S.toAddSubmonoid
      source
      @[simp]
      theorem Ideal.pointwise_smul_toAddSubGroup {M : Type u_1} [Monoid M] {R : Type u_3} [Ring R] [MulSemiringAction M R] (a : M) (S : Ideal R) (ha : Function.Surjective fun (r : R) => a • r) :
      Submodule.toAddSubgroup (a • S) = a • Submodule.toAddSubgroup S
      source
      theorem Ideal.pointwise_smul_eq_comap {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} (S : Ideal R) :
      a • S = Ideal.comap (RingEquiv.symm ((MulSemiringAction.toRingAut M R) a)) S
      source
      @[simp]
      theorem Ideal.smul_mem_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R} :
      a • x ∈ a • S ↔ x ∈ S
      source
      theorem Ideal.mem_pointwise_smul_iff_inv_smul_mem {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R} :
      x ∈ a • S ↔ a⁻¹ • x ∈ S
      source
      theorem Ideal.mem_inv_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Ideal R} {x : R} :
      x ∈ a⁻¹ • S ↔ a • x ∈ S
      source
      @[simp]
      theorem Ideal.pointwise_smul_le_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S T : Ideal R} :
      a • S ≤ a • T ↔ S ≤ T
      source
      theorem Ideal.pointwise_smul_subset_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S T : Ideal R} :
      a • S ≤ T ↔ S ≤ a⁻¹ • T
      source
      theorem Ideal.subset_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S T : Ideal R} :
      S ≤ a • T ↔ a⁻¹ • S ≤ T
      source
      instance Ideal.IsPrime.smul {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {I : Ideal R} [H : I.IsPrime] (g : M) :
      (g • I).IsPrime
      source
      @[simp]
      theorem Ideal.IsPrime.smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {I : Ideal R} (g : M) :
      (g • I).IsPrime ↔ I.IsPrime

      TODO: add equivSMul like we have for subgroup.