Lemmas about regular elements in rings.

theorem isLeftRegular_of_non_zero_divisor {α : Type u_1} [NonUnitalNonAssocRing α] (k : α) (h : ∀ (x : α), k * x = 0 → x = 0) : IsLeftRegular k

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} [NonUnitalNonAssocRing α] (k : α) (h : ∀ (x : α), x * k = 0 → x = 0) : IsRightRegular k

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} [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} (hk : k ≠ 0) : IsRegular k

theorem isRegular_iff_ne_zero' {α : Type u_1} [Nontrivial α] [NonUnitalNonAssocRing α] [NoZeroDivisors α] {k : α} : IsRegular k ↔ k ≠ 0

@[reducible]

def NoZeroDivisors.toCancelMonoidWithZero {α : Type u_1} [Ring α] [NoZeroDivisors α] : CancelMonoidWithZero α

A ring with no zero divisors is a CancelMonoidWithZero.

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

Equations

• NoZeroDivisors.toCancelMonoidWithZero = let __src := inferInstance; CancelMonoidWithZero.mk

@[reducible]

def NoZeroDivisors.toCancelCommMonoidWithZero {α : Type u_1} [CommRing α] [NoZeroDivisors α] : CancelCommMonoidWithZero α

A commutative ring with no zero divisors is a CancelCommMonoidWithZero.

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

Equations

• NoZeroDivisors.toCancelCommMonoidWithZero = let __src := NoZeroDivisors.toCancelMonoidWithZero; let __src_1 := inst✝; CancelCommMonoidWithZero.mk

instance IsDomain.toCancelMonoidWithZero {α : Type u_1} [Semiring α] [IsDomain α] : CancelMonoidWithZero α

Equations

• IsDomain.toCancelMonoidWithZero = CancelMonoidWithZero.mk

instance IsDomain.toCancelCommMonoidWithZero {α : Type u_1} [CommSemiring α] [IsDomain α] : CancelCommMonoidWithZero α

Equations

• IsDomain.toCancelCommMonoidWithZero = CancelCommMonoidWithZero.mk