Non-zero divisors in a ring #
theorem
IsLeftRegular.pow_injective
{R : Type u_1}
[Monoid R]
{r : R}
[IsMulTorsionFree R]
(hx : IsLeftRegular r)
(hx' : r ≠ 1)
:
Function.Injective fun (n : ℕ) => r ^ n
theorem
IsAddLeftRegular.nsmul_injective
{R : Type u_1}
[AddMonoid R]
{r : R}
[IsAddTorsionFree R]
(hx : IsAddLeftRegular r)
(hx' : r ≠ 0)
:
Function.Injective fun (n : ℕ) => n • r
theorem
IsRightRegular.pow_injective
{M : Type u_2}
[Monoid M]
[IsMulTorsionFree M]
{x : M}
(hx : IsRightRegular x)
(hx' : x ≠ 1)
:
Function.Injective fun (n : ℕ) => x ^ n
theorem
IsAddRightRegular.nsmul_injective
{M : Type u_2}
[AddMonoid M]
[IsAddTorsionFree M]
{x : M}
(hx : IsAddRightRegular x)
(hx' : x ≠ 0)
:
Function.Injective fun (n : ℕ) => n • x
theorem
IsMulTorsionFree.pow_right_injective
{M : Type u_2}
[CancelMonoid M]
[IsMulTorsionFree M]
{x : M}
(hx : x ≠ 1)
:
Function.Injective fun (n : ℕ) => x ^ n
@[simp]
theorem
IsMulTorsionFree.pow_right_inj
{M : Type u_2}
[CancelMonoid M]
[IsMulTorsionFree M]
{x : M}
(hx : x ≠ 1)
{n m : ℕ}
:
theorem
IsMulTorsionFree.pow_right_injective₀
{M : Type u_2}
[CancelMonoidWithZero M]
[IsMulTorsionFree M]
{x : M}
(hx : x ≠ 1)
(hx' : x ≠ 0)
:
Function.Injective fun (n : ℕ) => x ^ n
@[simp]
theorem
IsMulTorsionFree.pow_right_inj₀
{M : Type u_2}
[CancelMonoidWithZero M]
[IsMulTorsionFree M]
{x : M}
(hx : x ≠ 1)
(hx' : x ≠ 0)
{n m : ℕ}
:
theorem
IsLeftRegular.isUnit_of_finite
{R : Type u_1}
[Monoid R]
{r : R}
[Finite R]
(h : IsLeftRegular r)
:
IsUnit r
theorem
IsRightRegular.isUnit_of_finite
{R : Type u_1}
[Monoid R]
{r : R}
[Finite R]
(h : IsRightRegular r)
:
IsUnit r
@[deprecated isLeftRegular_iff_mem_nonZeroDivisorsLeft (since := "2025-07-16")]
@[deprecated isRightRegular_iff_mem_nonZeroDivisorsRight (since := "2025-07-16")]
@[deprecated le_nonZeroDivisorsRight_iff_isRightRegular (since := "2025-07-16")]
theorem
mul_cancel_left_coe_nonZeroDivisors
{R : Type u_1}
[Ring R]
{x y : R}
{c : ↥(nonZeroDivisors R)}
:
theorem
mul_cancel_right_coe_nonZeroDivisors
{R : Type u_1}
[Ring R]
{x y : R}
{c : ↥(nonZeroDivisors R)}
:
In a finite ring, an element is a unit iff it is a non-zero-divisor.
theorem
dvd_cancel_left_coe_nonZeroDivisors
{R : Type u_1}
[Ring R]
{x y : R}
{c : ↥(nonZeroDivisors R)}
:
theorem
dvd_cancel_right_coe_nonZeroDivisors
{R : Type u_1}
[CommRing R]
{x y : R}
{c : ↥(nonZeroDivisors R)}
: