Documentation

Mathlib.Algebra.Ring.Opposite

Ring structures on the multiplicative opposite #

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instance MulOpposite.ring (α : Type u) [Ring α] :
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instance AddOpposite.ring (α : Type u) [Ring α] :
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@[simp]
theorem NonUnitalRingHom.toOpposite_apply {R : Type u_1} {S : Type u_2} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
↑(NonUnitalRingHom.toOpposite f hf) = MulOpposite.op f
def NonUnitalRingHom.toOpposite {R : Type u_1} {S : Type u_2} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :

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.

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    @[simp]
    theorem NonUnitalRingHom.fromOpposite_apply {R : Type u_1} {S : Type u_2} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
    ↑(NonUnitalRingHom.fromOpposite f hf) = f MulOpposite.unop
    def NonUnitalRingHom.fromOpposite {R : Type u_1} {S : Type u_2} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R →ₙ+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :

    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.

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      @[simp]
      theorem NonUnitalRingHom.op_apply_apply {α : Type u_1} {β : Type u_2} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] (f : α →ₙ+* β) :
      ∀ (a : αᵐᵒᵖ), ↑(NonUnitalRingHom.op f) a = ZeroHom.toFun (↑(AddMonoidHom.mulOp (NonUnitalRingHom.toAddMonoidHom f))) a
      @[simp]
      theorem NonUnitalRingHom.op_symm_apply_apply {α : Type u_1} {β : Type u_2} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] (f : αᵐᵒᵖ →ₙ+* βᵐᵒᵖ) :
      ∀ (a : α), ↑(NonUnitalRingHom.op.symm f) a = ZeroHom.toFun (↑(AddMonoidHom.mulUnop (NonUnitalRingHom.toAddMonoidHom f))) a

      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.

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

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        • NonUnitalRingHom.unop = NonUnitalRingHom.op.symm
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          @[simp]
          theorem RingHom.toOpposite_apply {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
          ↑(RingHom.toOpposite f hf) = MulOpposite.op f
          def RingHom.toOpposite {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :

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

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            @[simp]
            theorem RingHom.fromOpposite_apply {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :
            ↑(RingHom.fromOpposite f hf) = f MulOpposite.unop
            def RingHom.fromOpposite {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) (hf : ∀ (x y : R), Commute (f x) (f y)) :

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

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              @[simp]
              theorem RingHom.op_symm_apply_apply {α : Type u_1} {β : Type u_2} [NonAssocSemiring α] [NonAssocSemiring β] (f : αᵐᵒᵖ →+* βᵐᵒᵖ) :
              ∀ (a : α), ↑(RingHom.op.symm f) a = MulOpposite.unop (f (MulOpposite.op a))
              @[simp]
              theorem RingHom.op_apply_apply {α : Type u_1} {β : Type u_2} [NonAssocSemiring α] [NonAssocSemiring β] (f : α →+* β) :
              ∀ (a : αᵐᵒᵖ), ↑(RingHom.op f) a = MulOpposite.op (f (MulOpposite.unop a))
              def RingHom.op {α : Type u_1} {β : Type u_2} [NonAssocSemiring α] [NonAssocSemiring β] :

              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.

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                def RingHom.unop {α : Type u_1} {β : Type u_2} [NonAssocSemiring α] [NonAssocSemiring β] :

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

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                • RingHom.unop = RingHom.op.symm
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