# Documentation

Mathlib.Algebra.Ring.Regular

# Lemmas about regular elements in rings. #

theorem isLeftRegular_of_non_zero_divisor {α : Type u_1} (k : α) (h : ∀ (x : α), k * x = 0x = 0) :

Left Mul by a k : α over [Ring α] is injective, if k is not a zero divisor. The typeclass that restricts all terms of α to have this property is NoZeroDivisors.

theorem isRightRegular_of_non_zero_divisor {α : Type u_1} (k : α) (h : ∀ (x : α), x * k = 0x = 0) :

Right Mul by a k : α over [Ring α] is injective, if k is not a zero divisor. The typeclass that restricts all terms of α to have this property is NoZeroDivisors.

theorem isRegular_of_ne_zero' {α : Type u_1} [] {k : α} (hk : k 0) :
theorem isRegular_iff_ne_zero' {α : Type u_1} [] [] {k : α} :
k 0
@[reducible]

A ring with no zero divisors is a CancelMonoidWithZero.

Note this is not an instance as it forms a typeclass loop.

Instances For
@[reducible]

A commutative ring with no zero divisors is a CancelCommMonoidWithZero.

Note this is not an instance as it forms a typeclass loop.

Instances For
instance IsDomain.toCancelMonoidWithZero {α : Type u_1} [] [] :