Documentation

Mathlib.Algebra.Ring.Idempotents

Idempotents #

This file defines idempotents for an arbitary multiplication and proves some basic results, including:

IsIdempotentElem.mul_of_commute: In a semigroup, the product of two commuting idempotents is an idempotent;
IsIdempotentElem.one_sub_iff: In a (non-associative) ring, p is an idempotent if and only if 1-p is an idempotent.
IsIdempotentElem.pow_succ_eq: In a monoid p ^ (n+1) = p for p an idempotent and n a natural number.

Tags #

projection, idempotent

def IsIdempotentElem {M : Type u_1} [inst : Mul M] (p : M) :

An element p is said to be idempotent if p * p = p

Equations

IsIdempotentElem p = (p * p = p)

theorem IsIdempotentElem.of_isIdempotent {M : Type u_1} [inst : Mul M] [inst : IsIdempotent M fun x x_1 => x * x_1] (a : M) :

IsIdempotentElem a

theorem IsIdempotentElem.eq {M : Type u_1} [inst : Mul M] {p : M} (h : IsIdempotentElem p) :

p * p = p

theorem IsIdempotentElem.mul_of_commute {S : Type u_1} [inst : Semigroup S] {p : S} {q : S} (h : Commute p q) (h₁ : IsIdempotentElem p) (h₂ : IsIdempotentElem q) :

IsIdempotentElem (p * q)

theorem IsIdempotentElem.zero {M₀ : Type u_1} [inst : MulZeroClass M₀] :

IsIdempotentElem 0

theorem IsIdempotentElem.one {M₁ : Type u_1} [inst : MulOneClass M₁] :

IsIdempotentElem 1

theorem IsIdempotentElem.one_sub {R : Type u_1} [inst : NonAssocRing R] {p : R} (h : IsIdempotentElem p) :

IsIdempotentElem (1 - p)

@[simp]

theorem IsIdempotentElem.one_sub_iff {R : Type u_1} [inst : NonAssocRing R] {p : R} :

IsIdempotentElem (1 - p) ↔ IsIdempotentElem p

theorem IsIdempotentElem.pow {N : Type u_1} [inst : Monoid N] {p : N} (n : ℕ) (h : IsIdempotentElem p) :

IsIdempotentElem (p ^ n)

theorem IsIdempotentElem.pow_succ_eq {N : Type u_1} [inst : Monoid N] {p : N} (n : ℕ) (h : IsIdempotentElem p) :

p ^ (n + 1) = p

@[simp]

theorem IsIdempotentElem.iff_eq_one {G : Type u_1} [inst : Group G] {p : G} :

IsIdempotentElem p ↔ p = 1

@[simp]

theorem IsIdempotentElem.iff_eq_zero_or_one {G₀ : Type u_1} [inst : CancelMonoidWithZero G₀] {p : G₀} :

IsIdempotentElem p ↔ p = 0 ∨ p = 1

Instances on `Subtype IsIdempotentElem` #

instance IsIdempotentElem.instZeroSubtypeIsIdempotentElemToMul {M₀ : Type u_1} [inst : MulZeroClass M₀] :

Zero { p // IsIdempotentElem p }

Equations

IsIdempotentElem.instZeroSubtypeIsIdempotentElemToMul = { zero := { val := 0, property := (_ : IsIdempotentElem 0) } }

@[simp]

theorem IsIdempotentElem.coe_zero {M₀ : Type u_1} [inst : MulZeroClass M₀] :

↑0 = 0

instance IsIdempotentElem.instOneSubtypeIsIdempotentElemToMul {M₁ : Type u_1} [inst : MulOneClass M₁] :

One { p // IsIdempotentElem p }

Equations

IsIdempotentElem.instOneSubtypeIsIdempotentElemToMul = { one := { val := 1, property := (_ : IsIdempotentElem 1) } }

@[simp]

theorem IsIdempotentElem.coe_one {M₁ : Type u_1} [inst : MulOneClass M₁] :

↑1 = 1

instance IsIdempotentElem.instHasComplSubtypeIsIdempotentElemToMulToNonUnitalNonAssocRing {R : Type u_1} [inst : NonAssocRing R] :

HasCompl { p // IsIdempotentElem p }

Equations

IsIdempotentElem.instHasComplSubtypeIsIdempotentElemToMulToNonUnitalNonAssocRing = { compl := fun p => { val := 1 - ↑p, property := (_ : IsIdempotentElem (1 - ↑p)) } }

@[simp]

theorem IsIdempotentElem.coe_compl {R : Type u_1} [inst : NonAssocRing R] (p : { p // IsIdempotentElem p }) :

↑(pᶜ) = 1 - ↑p

@[simp]

theorem IsIdempotentElem.compl_compl {R : Type u_1} [inst : NonAssocRing R] (p : { p // IsIdempotentElem p }) :

@[simp]

theorem IsIdempotentElem.zero_compl {R : Type u_1} [inst : NonAssocRing R] :

0ᶜ = 1

@[simp]

theorem IsIdempotentElem.one_compl {R : Type u_1} [inst : NonAssocRing R] :

1ᶜ = 0