Idempotent elements of a ring

This file proves result about idempotent elements of a ring, like:

• IsIdempotentElem.one_sub_iff: In a (non-associative) ring, a is an idempotent if and only if 1 - a is an idempotent.

theorem IsIdempotentElem.one_sub {R : Type u_1} [NonAssocRing R] {a : R} (h : IsIdempotentElem a) : IsIdempotentElem (1 - a)

@[simp]

theorem IsIdempotentElem.one_sub_iff {R : Type u_1} [NonAssocRing R] {a : R} : IsIdempotentElem (1 - a) ↔ IsIdempotentElem a

@[simp]

theorem IsIdempotentElem.mul_one_sub_self {R : Type u_1} [NonAssocRing R] {a : R} (h : IsIdempotentElem a) : a * (1 - a) = 0

@[simp]

theorem IsIdempotentElem.one_sub_mul_self {R : Type u_1} [NonAssocRing R] {a : R} (h : IsIdempotentElem a) : (1 - a) * a = 0

instance IsIdempotentElem.instHasComplSubtype {R : Type u_1} [NonAssocRing R] : HasCompl { a : R // IsIdempotentElem a }

Equations

• IsIdempotentElem.instHasComplSubtype = { compl := fun (a : { a : R // IsIdempotentElem a }) => ⟨1 - ↑a, ⋯⟩ }

@[simp]

theorem IsIdempotentElem.coe_compl {R : Type u_1} [NonAssocRing R] (a : { a : R // IsIdempotentElem a }) : ↑aᶜ = 1 - ↑a

@[simp]

theorem IsIdempotentElem.compl_compl {R : Type u_1} [NonAssocRing R] (a : { a : R // IsIdempotentElem a }) : aᶜᶜ = a

@[simp]

theorem IsIdempotentElem.zero_compl {R : Type u_1} [NonAssocRing R] : 0ᶜ = 1

@[simp]

theorem IsIdempotentElem.one_compl {R : Type u_1} [NonAssocRing R] : 1ᶜ = 0

theorem IsIdempotentElem.of_mul_add {R : Type u_1} [Semiring R] {a b : R} (mul : a * b = 0) (add : a + b = 1) : IsIdempotentElem a ∧ IsIdempotentElem b

theorem IsIdempotentElem.add_sub_mul_of_commute {R : Type u_1} [Ring R] {a b : R} (h : Commute a b) (hp : IsIdempotentElem a) (hq : IsIdempotentElem b) : IsIdempotentElem (a + b - a * b)

theorem IsIdempotentElem.add_sub_mul {R : Type u_1} [CommRing R] {a b : R} (hp : IsIdempotentElem a) (hq : IsIdempotentElem b) : IsIdempotentElem (a + b - a * b)