mathlib documentation

algebra.char_zero

Characteristic zero #

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 definition #

char_zero is the typeclass of an additive monoid with one such that the natural homomorphism from the natural numbers into it is injective.

Main statements #

TODO #

@[class]
structure char_zero (R : Type u_1) [add_monoid R] [has_one R] :
Prop

Typeclass for monoids with characteristic zero. (This is usually stated on fields but it makes sense for any additive monoid with 1.)

Instances
theorem char_zero_of_inj_zero {R : Type u_1} [add_left_cancel_monoid R] [has_one R] (H : ∀ (n : ), n = 0n = 0) :

Note this is not an instance as char_zero implies nontrivial, and this would risk forming a loop.

@[simp]
theorem nat.cast_inj {R : Type u_1} [add_monoid R] [has_one R] [char_zero R] {m n : } :
m = n m = n
@[simp]
theorem nat.cast_eq_zero {R : Type u_1} [add_monoid R] [has_one R] [char_zero R] {n : } :
n = 0 n = 0
theorem nat.cast_ne_zero {R : Type u_1} [add_monoid R] [has_one R] [char_zero R] {n : } :
n 0 n 0
theorem nat.cast_add_one_ne_zero {R : Type u_1} [add_monoid R] [has_one R] [char_zero R] (n : ) :
n + 1 0
@[simp]
theorem nat.cast_dvd_char_zero {k : Type u_1} [field k] [char_zero k] {m n : } (n_dvd : n m) :
(m / n) = m / n
@[instance]
def char_zero.infinite (M : Type u_1) [add_monoid M] [has_one M] [char_zero M] :
theorem two_ne_zero' {M : Type u_1} [add_monoid M] [has_one M] [char_zero M] :
2 0
@[simp]
theorem add_self_eq_zero {R : Type u_1} [semiring R] [no_zero_divisors R] [char_zero R] {a : R} :
a + a = 0 a = 0
@[simp]
theorem bit0_eq_zero {R : Type u_1} [semiring R] [no_zero_divisors R] [char_zero R] {a : R} :
bit0 a = 0 a = 0
@[simp]
theorem zero_eq_bit0 {R : Type u_1} [semiring R] [no_zero_divisors R] [char_zero R] {a : R} :
0 = bit0 a a = 0
theorem neg_eq_self_iff {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a : R} :
-a = a a = 0
theorem eq_neg_self_iff {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a : R} :
a = -a a = 0
theorem nat_mul_inj {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {n : } {a b : R} (h : (n) * a = (n) * b) :
n = 0 a = b
theorem nat_mul_inj' {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {n : } {a b : R} (h : (n) * a = (n) * b) (w : n 0) :
a = b
@[simp]
theorem bit0_eq_bit0 {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a b : R} :
bit0 a = bit0 b a = b
@[simp]
theorem bit1_eq_bit1 {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a b : R} :
bit1 a = bit1 b a = b
@[simp]
theorem bit1_eq_one {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a : R} :
bit1 a = 1 a = 0
@[simp]
theorem one_eq_bit1 {R : Type u_1} [ring R] [no_zero_divisors R] [char_zero R] {a : R} :
1 = bit1 a a = 0
@[simp]
theorem half_add_self {R : Type u_1} [division_ring R] [char_zero R] (a : R) :
(a + a) / 2 = a
@[simp]
theorem add_halves' {R : Type u_1} [division_ring R] [char_zero R] (a : R) :
a / 2 + a / 2 = a
theorem sub_half {R : Type u_1} [division_ring R] [char_zero R] (a : R) :
a - a / 2 = a / 2
theorem half_sub {R : Type u_1} [division_ring R] [char_zero R] (a : R) :
a / 2 - a = -(a / 2)
@[instance]
def with_top.char_zero {R : Type u_1} [add_monoid R] [has_one R] [char_zero R] :
theorem ring_hom.char_zero {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (ϕ : R →+* S) [hS : char_zero S] :
theorem ring_hom.char_zero_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] {ϕ : R →+* S} (hϕ : function.injective ϕ) :