# Characteristic zero (additional theorems) #

A ring R is called of characteristic zero if every natural number n is non-zero when considered as an element of R. Since this definition doesn't mention the multiplicative structure of R except for the existence of 1 in this file characteristic zero is defined for additive monoids with 1.

## Main statements #

• Characteristic zero implies that the additive monoid is infinite.
@[simp]
theorem Nat.castEmbedding_apply {R : Type u_1} [] [] :
∀ (a : ), Nat.castEmbedding a = a
def Nat.castEmbedding {R : Type u_1} [] [] :

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

Equations
• Nat.castEmbedding = { toFun := Nat.cast, inj' := }
Instances For
@[simp]
theorem Nat.cast_pow_eq_one {R : Type u_2} [] [] (q : ) (n : ) (hn : n 0) :
q ^ n = 1 q = 1
@[simp]
theorem Nat.cast_div_charZero {k : Type u_2} [] [] {m : } {n : } (n_dvd : n m) :
(m / n) = m / n
instance CharZero.NeZero.two {M : Type u_2} [] [] :
Equations
• =
theorem Function.support_natCast {α : Type u_1} {M : Type u_2} [] [] {n : } (hn : n 0) :
= Set.univ
theorem Function.mulSupport_natCast {α : Type u_1} {M : Type u_2} [] [] {n : } (hn : n 1) :
= Set.univ
@[simp]
theorem add_self_eq_zero {R : Type u_1} [] [] [] {a : R} :
a + a = 0 a = 0
@[simp]
theorem bit0_eq_zero {R : Type u_1} [] [] [] {a : R} :
bit0 a = 0 a = 0
@[simp]
theorem zero_eq_bit0 {R : Type u_1} [] [] [] {a : R} :
0 = bit0 a a = 0
theorem bit0_ne_zero {R : Type u_1} [] [] [] {a : R} :
bit0 a 0 a 0
theorem zero_ne_bit0 {R : Type u_1} [] [] [] {a : R} :
0 bit0 a a 0
@[simp]
theorem neg_eq_self_iff {R : Type u_1} [] [] [] {a : R} :
-a = a a = 0
@[simp]
theorem eq_neg_self_iff {R : Type u_1} [] [] [] {a : R} :
a = -a a = 0
theorem nat_mul_inj {R : Type u_1} [] [] [] {n : } {a : R} {b : R} (h : n * a = n * b) :
n = 0 a = b
theorem nat_mul_inj' {R : Type u_1} [] [] [] {n : } {a : R} {b : R} (h : n * a = n * b) (w : n 0) :
a = b
theorem bit0_injective {R : Type u_1} [] [] [] :
theorem bit1_injective {R : Type u_1} [] [] [] :
@[simp]
theorem bit0_eq_bit0 {R : Type u_1} [] [] [] {a : R} {b : R} :
bit0 a = bit0 b a = b
@[simp]
theorem bit1_eq_bit1 {R : Type u_1} [] [] [] {a : R} {b : R} :
bit1 a = bit1 b a = b
@[simp]
theorem bit1_eq_one {R : Type u_1} [] [] [] {a : R} :
bit1 a = 1 a = 0
@[simp]
theorem one_eq_bit1 {R : Type u_1} [] [] [] {a : R} :
1 = bit1 a a = 0
@[simp]
theorem half_add_self {R : Type u_1} [] [] (a : R) :
(a + a) / 2 = a
@[simp]
theorem add_halves' {R : Type u_1} [] [] (a : R) :
a / 2 + a / 2 = a
theorem sub_half {R : Type u_1} [] [] (a : R) :
a - a / 2 = a / 2
theorem half_sub {R : Type u_1} [] [] (a : R) :
a / 2 - a = -(a / 2)
Equations
• =
Equations
• =
theorem RingHom.charZero {R : Type u_1} {S : Type u_2} [] [] (ϕ : R →+* S) [hS : ] :
theorem RingHom.charZero_iff {R : Type u_1} {S : Type u_2} [] [] {ϕ : R →+* S} (hϕ : ) :
theorem RingHom.injective_nat {R : Type u_1} [] (f : →+* R) [] :
@[simp]
theorem units_ne_neg_self {R : Type u_1} [Ring R] [] (u : Rˣ) :
u -u
@[simp]
theorem neg_units_ne_self {R : Type u_1} [Ring R] [] (u : Rˣ) :
-u u