Documentation

Mathlib.Algebra.Ring.Opposite

Ring structures on the multiplicative opposite #

instance MulOpposite.instRing {α : Type u_1} [Ring α] :
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instance AddOpposite.instRing {α : Type u_1} [Ring α] :
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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)) :

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.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)) :
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)) :

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.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)) :

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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  • One or more equations did not get rendered due to their size.
@[simp]
theorem NonUnitalRingHom.op_apply_apply {α : Type u_2} {β : Type u_3} [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] (f : α →ₙ+* β) (a✝ : αᵐᵒᵖ) :
(op f) a✝ = (↑(AddMonoidHom.mulOp f.toAddMonoidHom)).toFun a✝

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

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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)) :

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.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)) :
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)) :

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

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

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