Documentation

Mathlib.Algebra.Ring.Opposite

Ring structures on the multiplicative opposite #

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instance MulOpposite.instDistrib {α : Type u_1} [Distrib α] :
Distrib αᵐᵒᵖ
Equations
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instance MulOpposite.instMulZeroClass {α : Type u_1} [MulZeroClass α] :
MulZeroClass αᵐᵒᵖ
Equations
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instance MulOpposite.instMulZeroOneClass {α : Type u_1} [MulZeroOneClass α] :
MulZeroOneClass αᵐᵒᵖ
Equations
  • MulOpposite.instMulZeroOneClass = let __spread.0 := MulOpposite.instMulOneClass; let __spread.1 := MulOpposite.instMulZeroClass; MulZeroOneClass.mk ⋯ ⋯
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instance MulOpposite.instSemigroupWithZero {α : Type u_1} [SemigroupWithZero α] :
SemigroupWithZero αᵐᵒᵖ
Equations
  • MulOpposite.instSemigroupWithZero = let __spread.0 := MulOpposite.instSemigroup; let __spread.1 := MulOpposite.instMulZeroClass; SemigroupWithZero.mk ⋯ ⋯
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instance MulOpposite.instMonoidWithZero {α : Type u_1} [MonoidWithZero α] :
MonoidWithZero αᵐᵒᵖ
Equations
  • MulOpposite.instMonoidWithZero = let __spread.0 := MulOpposite.instMonoid; let __spread.1 := MulOpposite.instMulZeroOneClass; MonoidWithZero.mk ⋯ ⋯
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instance MulOpposite.instNonUnitalNonAssocSemiring {α : Type u_1} [NonUnitalNonAssocSemiring α] :
NonUnitalNonAssocSemiring αᵐᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance MulOpposite.instNonUnitalSemiring {α : Type u_1} [NonUnitalSemiring α] :
NonUnitalSemiring αᵐᵒᵖ
Equations
  • MulOpposite.instNonUnitalSemiring = let __spread.0 := MulOpposite.instNonUnitalNonAssocSemiring; let __spread.1 := MulOpposite.instSemigroupWithZero; NonUnitalSemiring.mk ⋯
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instance MulOpposite.instNonAssocSemiring {α : Type u_1} [NonAssocSemiring α] :
NonAssocSemiring αᵐᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance MulOpposite.instSemiring {α : Type u_1} [Semiring α] :
Semiring αᵐᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance MulOpposite.instNonUnitalCommSemiring {α : Type u_1} [NonUnitalCommSemiring α] :
NonUnitalCommSemiring αᵐᵒᵖ
Equations
  • MulOpposite.instNonUnitalCommSemiring = let __spread.0 := MulOpposite.instNonUnitalSemiring; let __spread.1 := MulOpposite.instCommSemigroup; NonUnitalCommSemiring.mk ⋯
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instance MulOpposite.instCommSemiring {α : Type u_1} [CommSemiring α] :
CommSemiring αᵐᵒᵖ
Equations
  • MulOpposite.instCommSemiring = let __spread.0 := MulOpposite.instSemiring; let __spread.1 := MulOpposite.instCommMonoid; CommSemiring.mk ⋯
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instance MulOpposite.instNonUnitalNonAssocRing {α : Type u_1} [NonUnitalNonAssocRing α] :
NonUnitalNonAssocRing αᵐᵒᵖ
Equations
  • MulOpposite.instNonUnitalNonAssocRing = let __spread.0 := MulOpposite.instAddCommGroup; let __spread.1 := MulOpposite.instNonUnitalNonAssocSemiring; NonUnitalNonAssocRing.mk ⋯ ⋯ ⋯ ⋯
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instance MulOpposite.instNonUnitalRing {α : Type u_1} [NonUnitalRing α] :
NonUnitalRing αᵐᵒᵖ
Equations
  • MulOpposite.instNonUnitalRing = let __spread.0 := MulOpposite.instNonUnitalNonAssocRing; let __spread.1 := MulOpposite.instNonUnitalSemiring; NonUnitalRing.mk ⋯
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instance MulOpposite.instNonAssocRing {α : Type u_1} [NonAssocRing α] :
NonAssocRing αᵐᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance MulOpposite.instRing {α : Type u_1} [Ring α] :
Ring αᵐᵒᵖ
Equations
  • MulOpposite.instRing = let __spread.0 := MulOpposite.instSemiring; let __spread.1 := MulOpposite.instAddCommGroupWithOne; Ring.mk ⋯ SubNegMonoid.zsmul ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
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instance MulOpposite.instNonUnitalCommRing {α : Type u_1} [NonUnitalCommRing α] :
NonUnitalCommRing αᵐᵒᵖ
Equations
  • MulOpposite.instNonUnitalCommRing = let __spread.0 := MulOpposite.instNonUnitalRing; let __spread.1 := MulOpposite.instNonUnitalCommSemiring; NonUnitalCommRing.mk ⋯
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instance MulOpposite.instCommRing {α : Type u_1} [CommRing α] :
CommRing αᵐᵒᵖ
Equations
  • MulOpposite.instCommRing = let __spread.0 := MulOpposite.instRing; let __spread.1 := MulOpposite.instCommMonoid; CommRing.mk ⋯
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instance MulOpposite.instNoZeroDivisors {α : Type u_1} [Zero α] [Mul α] [NoZeroDivisors α] :
NoZeroDivisors αᵐᵒᵖ
Equations
  • ⋯ = ⋯
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instance MulOpposite.instIsDomain {α : Type u_1} [Ring α] [IsDomain α] :
IsDomain αᵐᵒᵖ
Equations
  • ⋯ = ⋯
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instance MulOpposite.instGroupWithZero {α : Type u_1} [GroupWithZero α] :
GroupWithZero αᵐᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance AddOpposite.instDistrib {α : Type u_1} [Distrib α] :
Distrib αᵃᵒᵖ
Equations
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instance AddOpposite.instMulZeroClass {α : Type u_1} [MulZeroClass α] :
MulZeroClass αᵃᵒᵖ
Equations
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instance AddOpposite.instMulZeroOneClass {α : Type u_1} [MulZeroOneClass α] :
MulZeroOneClass αᵃᵒᵖ
Equations
  • AddOpposite.instMulZeroOneClass = let __spread.0 := AddOpposite.instMulOneClass; let __spread.1 := AddOpposite.instMulZeroClass; MulZeroOneClass.mk ⋯ ⋯
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instance AddOpposite.instSemigroupWithZero {α : Type u_1} [SemigroupWithZero α] :
SemigroupWithZero αᵃᵒᵖ
Equations
  • AddOpposite.instSemigroupWithZero = let __spread.0 := AddOpposite.instSemigroup; let __spread.1 := AddOpposite.instMulZeroClass; SemigroupWithZero.mk ⋯ ⋯
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instance AddOpposite.instMonoidWithZero {α : Type u_1} [MonoidWithZero α] :
MonoidWithZero αᵃᵒᵖ
Equations
  • AddOpposite.instMonoidWithZero = let __spread.0 := AddOpposite.instMonoid; let __spread.1 := AddOpposite.instMulZeroOneClass; MonoidWithZero.mk ⋯ ⋯
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instance AddOpposite.instNonUnitalNonAssocSemiring {α : Type u_1} [NonUnitalNonAssocSemiring α] :
NonUnitalNonAssocSemiring αᵃᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance AddOpposite.instNonUnitalSemiring {α : Type u_1} [NonUnitalSemiring α] :
NonUnitalSemiring αᵃᵒᵖ
Equations
  • AddOpposite.instNonUnitalSemiring = let __spread.0 := AddOpposite.instNonUnitalNonAssocSemiring; let __spread.1 := AddOpposite.instSemigroupWithZero; NonUnitalSemiring.mk ⋯
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instance AddOpposite.instNonAssocSemiring {α : Type u_1} [NonAssocSemiring α] :
NonAssocSemiring αᵃᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance AddOpposite.instSemiring {α : Type u_1} [Semiring α] :
Semiring αᵃᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance AddOpposite.instNonUnitalCommSemiring {α : Type u_1} [NonUnitalCommSemiring α] :
NonUnitalCommSemiring αᵃᵒᵖ
Equations
  • AddOpposite.instNonUnitalCommSemiring = let __spread.0 := AddOpposite.instNonUnitalSemiring; let __spread.1 := AddOpposite.instCommSemigroup; NonUnitalCommSemiring.mk ⋯
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instance AddOpposite.instCommSemiring {α : Type u_1} [CommSemiring α] :
CommSemiring αᵃᵒᵖ
Equations
  • AddOpposite.instCommSemiring = let __spread.0 := AddOpposite.instSemiring; let __spread.1 := AddOpposite.instCommMonoid; CommSemiring.mk ⋯
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instance AddOpposite.instNonUnitalNonAssocRing {α : Type u_1} [NonUnitalNonAssocRing α] :
NonUnitalNonAssocRing αᵃᵒᵖ
Equations
  • AddOpposite.instNonUnitalNonAssocRing = let __spread.0 := AddOpposite.instAddCommGroup; let __spread.1 := AddOpposite.instNonUnitalNonAssocSemiring; NonUnitalNonAssocRing.mk ⋯ ⋯ ⋯ ⋯
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instance AddOpposite.instNonUnitalRing {α : Type u_1} [NonUnitalRing α] :
NonUnitalRing αᵃᵒᵖ
Equations
  • AddOpposite.instNonUnitalRing = let __spread.0 := AddOpposite.instNonUnitalNonAssocRing; let __spread.1 := AddOpposite.instNonUnitalSemiring; NonUnitalRing.mk ⋯
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instance AddOpposite.instNonAssocRing {α : Type u_1} [NonAssocRing α] :
NonAssocRing αᵃᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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instance AddOpposite.instRing {α : Type u_1} [Ring α] :
Ring αᵃᵒᵖ
Equations
  • AddOpposite.instRing = let __spread.0 := AddOpposite.instSemiring; let __spread.1 := AddOpposite.instAddCommGroupWithOne; Ring.mk ⋯ SubNegMonoid.zsmul ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
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instance AddOpposite.instNonUnitalCommRing {α : Type u_1} [NonUnitalCommRing α] :
NonUnitalCommRing αᵃᵒᵖ
Equations
  • AddOpposite.instNonUnitalCommRing = let __spread.0 := AddOpposite.instNonUnitalRing; let __spread.1 := AddOpposite.instNonUnitalCommSemiring; NonUnitalCommRing.mk ⋯
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instance AddOpposite.instCommRing {α : Type u_1} [CommRing α] :
CommRing αᵃᵒᵖ
Equations
  • AddOpposite.instCommRing = let __spread.0 := AddOpposite.instRing; let __spread.1 := AddOpposite.instCommMonoid; CommRing.mk ⋯
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instance AddOpposite.instNoZeroDivisors {α : Type u_1} [Zero α] [Mul α] [NoZeroDivisors α] :
NoZeroDivisors αᵃᵒᵖ
Equations
  • ⋯ = ⋯
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instance AddOpposite.instIsDomain {α : Type u_1} [Ring α] [IsDomain α] :
IsDomain αᵃᵒᵖ
Equations
  • ⋯ = ⋯
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instance AddOpposite.instGroupWithZero {α : Type u_1} [GroupWithZero α] :
GroupWithZero αᵃᵒᵖ
Equations
  • One or more equations did not get rendered due to their size.
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@[simp]
theorem NonUnitalRingHom.toOpposite_apply {R : Type u_2} {S : Type u_3} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
⇑(NonUnitalRingHom.toOpposite f hf) = MulOpposite.op ∘ ⇑f
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def NonUnitalRingHom.toOpposite {R : Type u_2} {S : Type u_3} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
R →ₙ+* Sᵐᵒᵖ

A non-unital ring homomorphism f : R →ₙ+* S such that f x commutes with f y for all x, y defines a non-unital ring homomorphism to Sᵐᵒᵖ.

Equations
  • One or more equations did not get rendered due to their size.
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    @[simp]
    theorem NonUnitalRingHom.fromOpposite_apply {R : Type u_2} {S : Type u_3} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
    ⇑(NonUnitalRingHom.fromOpposite f hf) = ⇑f ∘ MulOpposite.unop
    source
    def NonUnitalRingHom.fromOpposite {R : Type u_2} {S : Type u_3} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
    Rᵐᵒᵖ →ₙ+* S

    A non-unital ring homomorphism f : R →ₙ* S such that f x commutes with f y for all x, y defines a non-unital ring homomorphism from Rᵐᵒᵖ.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
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      @[simp]
      theorem NonUnitalRingHom.op_symm_apply_apply {α : Type u_2} {β : Type u_3} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] (f : αᵐᵒᵖ →ₙ+* βᵐᵒᵖ) :
      ∀ (a : α), (NonUnitalRingHom.op.symm f) a = (AddMonoidHom.mulUnop (NonUnitalRingHom.toAddMonoidHom f)).toFun a
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      @[simp]
      theorem NonUnitalRingHom.op_apply_apply {α : Type u_2} {β : Type u_3} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] (f : α →ₙ+* β) :
      ∀ (a : αᵐᵒᵖ), (NonUnitalRingHom.op f) a = (AddMonoidHom.mulOp (NonUnitalRingHom.toAddMonoidHom f)).toFun a
      source
      def NonUnitalRingHom.op {α : Type u_2} {β : Type u_3} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] :
      (α →ₙ+* β) ≃ (αᵐᵒᵖ →ₙ+* βᵐᵒᵖ)

      A non-unital ring hom α →ₙ+* β can equivalently be viewed as a non-unital ring hom αᵐᵒᵖ →+* βᵐᵒᵖ. This is the action of the (fully faithful) ᵐᵒᵖ-functor on morphisms.

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
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        def NonUnitalRingHom.unop {α : Type u_2} {β : Type u_3} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] :
        (αᵐᵒᵖ →ₙ+* βᵐᵒᵖ) ≃ (α →ₙ+* β)

        The 'unopposite' of a non-unital ring hom αᵐᵒᵖ →ₙ+* βᵐᵒᵖ. Inverse to NonUnitalRingHom.op.

        Equations
        • NonUnitalRingHom.unop = NonUnitalRingHom.op.symm
        Instances For
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          @[simp]
          theorem RingHom.toOpposite_apply {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
          ⇑(RingHom.toOpposite f hf) = MulOpposite.op ∘ ⇑f
          source
          def RingHom.toOpposite {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
          R →+* Sᵐᵒᵖ

          A ring homomorphism f : R →+* S such that f x commutes with f y for all x, y defines a ring homomorphism to Sᵐᵒᵖ.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
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            @[simp]
            theorem RingHom.fromOpposite_apply {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
            ⇑(RingHom.fromOpposite f hf) = ⇑f ∘ MulOpposite.unop
            source
            def RingHom.fromOpposite {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
            Rᵐᵒᵖ →+* S

            A ring homomorphism f : R →+* S such that f x commutes with f y for all x, y defines a ring homomorphism from Rᵐᵒᵖ.

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
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              @[simp]
              theorem RingHom.op_apply_apply {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [NonAssocSemiring β] (f : α →+* β) :
              ∀ (a : αᵐᵒᵖ), (RingHom.op f) a = MulOpposite.op (f (MulOpposite.unop a))
              source
              @[simp]
              theorem RingHom.op_symm_apply_apply {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [NonAssocSemiring β] (f : αᵐᵒᵖ →+* βᵐᵒᵖ) :
              ∀ (a : α), (RingHom.op.symm f) a = MulOpposite.unop (f (MulOpposite.op a))
              source
              def RingHom.op {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [NonAssocSemiring β] :
              (α →+* β) ≃ (αᵐᵒᵖ →+* βᵐᵒᵖ)

              A ring hom α →+* β can equivalently be viewed as a ring hom αᵐᵒᵖ →+* βᵐᵒᵖ. This is the action of the (fully faithful) ᵐᵒᵖ-functor on morphisms.

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                source
                def RingHom.unop {α : Type u_2} {β : Type u_3} [NonAssocSemiring α] [NonAssocSemiring β] :
                (αᵐᵒᵖ →+* βᵐᵒᵖ) ≃ (α →+* β)

                The 'unopposite' of a ring hom αᵐᵒᵖ →+* βᵐᵒᵖ. Inverse to RingHom.op.

                Equations
                • RingHom.unop = RingHom.op.symm
                Instances For