Documentation

Mathlib.Algebra.Ring.CharZero

Characteristic zero rings #

source
def Nat.castEmbedding {R : Type u_2} [AddMonoidWithOne R] [CharZero R] :
ℕ ↪ R

Nat.cast as an embedding into monoids of characteristic 0.

Equations
Instances For
    source
    @[simp]
    theorem Nat.castEmbedding_apply {R : Type u_2} [AddMonoidWithOne R] [CharZero R] (a✝ : ℕ) :
    castEmbedding a✝ = ↑a✝
    source
    instance CharZero.NeZero.two {R : Type u_2} [AddMonoidWithOne R] [CharZero R] :
    NeZero 2
    source
    theorem Function.support_natCast {α : Type u_1} {R : Type u_2} {n : ℕ} [AddMonoidWithOne R] [CharZero R] (hn : n ≠ 0) :
    support ↑n = Set.univ
    source
    theorem Function.mulSupport_natCast {α : Type u_1} {R : Type u_2} {n : ℕ} [AddMonoidWithOne R] [CharZero R] (hn : n ≠ 1) :
    mulSupport ↑n = Set.univ
    source
    theorem RingHom.charZero {R : Type u_2} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] (ϕ : R →+* S) [CharZero S] :
    CharZero R
    source
    theorem RingHom.charZero_iff {R : Type u_2} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] {ϕ : R →+* S} (hϕ : Function.Injective ⇑ϕ) :
    CharZero R ↔ CharZero S
    source
    theorem RingHom.injective_nat {R : Type u_2} [NonAssocSemiring R] (f : ℕ →+* R) [CharZero R] :
    Function.Injective ⇑f
    source
    @[simp]
    theorem add_self_eq_zero {R : Type u_2} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R} :
    a + a = 0 ↔ a = 0
    source
    @[simp]
    theorem Nat.cast_pow_eq_one {R : Type u_2} {n : ℕ} [Semiring R] [CharZero R] {a : ℕ} (hn : n ≠ 0) :
    ↑a ^ n = 1 ↔ a = 1
    source
    @[instance 100]
    instance HasUniqueDiv.of_isCancelMulZero_charZero {R : Type u_2} [Semiring R] [CharZero R] [IsCancelMulZero R] :
    HasUniqueDiv R

    A characteristic zero domain is torsion-free.

    source
    theorem CharZero.neg_eq_self_iff {R : Type u_2} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    -a = a ↔ a = 0
    source
    theorem CharZero.eq_neg_self_iff {R : Type u_2} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {a : R} :
    a = -a ↔ a = 0
    source
    theorem nat_mul_inj {R : Type u_2} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {n : ℕ} {a b : R} (h : ↑n * a = ↑n * b) :
    n = 0 ∨ a = b
    source
    theorem nat_mul_inj' {R : Type u_2} [NonAssocRing R] [NoZeroDivisors R] [CharZero R] {n : ℕ} {a b : R} (h : ↑n * a = ↑n * b) (w : n ≠ 0) :
    a = b
    source
    @[simp]
    theorem units_ne_neg_self {R : Type u_2} [Ring R] [CharZero R] (u : Rˣ) :
    u ≠ -u
    source
    @[simp]
    theorem neg_units_ne_self {R : Type u_2} [Ring R] [CharZero R] (u : Rˣ) :
    -u ≠ u