Regular elements

We introduce left-regular, right-regular and regular elements, along with their to_additive analogues add-left-regular, add-right-regular and add-regular elements.

By definition, a regular element in a commutative ring is a non-zero divisor. Lemma isRegular_of_ne_zero implies that every non-zero element of an integral domain is regular. Since it assumes that the ring is a CancelMonoidWithZero it applies also, for instance, to ℕ.

The lemmas in Section MulZeroClass show that the 0 element is (left/right-)regular if and only if the MulZeroClass is trivial. This is useful when figuring out stopping conditions for regular sequences: if 0 is ever an element of a regular sequence, then we can extend the sequence by adding one further 0.

The final goal is to develop part of the API to prove, eventually, results about non-zero-divisors.

def IsLeftRegular {R : Type u_1} [Mul R] (c : R) : Prop

A left-regular element is an element c such that multiplication on the left by c is injective.

Equations

• IsLeftRegular c = Function.Injective fun (x : R) => c * x

def IsAddLeftRegular {R : Type u_1} [Add R] (c : R) : Prop

An add-left-regular element is an element c such that addition on the left by c is injective.

Equations

• IsAddLeftRegular c = Function.Injective fun (x : R) => c + x

def IsRightRegular {R : Type u_1} [Mul R] (c : R) : Prop

A right-regular element is an element c such that multiplication on the right by c is injective.

Equations

• IsRightRegular c = Function.Injective fun (x : R) => x * c

def IsAddRightRegular {R : Type u_1} [Add R] (c : R) : Prop

An add-right-regular element is an element c such that addition on the right by c is injective.

Equations

• IsAddRightRegular c = Function.Injective fun (x : R) => x + c

structure IsAddRegular {R : Type u_2} [Add R] (c : R) : Prop

An add-regular element is an element c such that addition by c both on the left and on the right is injective.

• left : IsAddLeftRegular c

  An add-regular element c is left-regular

• right : IsAddRightRegular c

  An add-regular element c is right-regular

structure IsRegular {R : Type u_1} [Mul R] (c : R) : Prop

A regular element is an element c such that multiplication by c both on the left and on the right is injective.

• left : IsLeftRegular c

  A regular element c is left-regular

• right : IsRightRegular c

  A regular element c is right-regular

theorem isRegular_iff {R : Type u_1} [Mul R] {c : R} : IsRegular c ↔ IsLeftRegular c ∧ IsRightRegular c

theorem isAddRegular_iff {R : Type u_1} [Add R] {c : R} : IsAddRegular c ↔ IsAddLeftRegular c ∧ IsAddRightRegular c

theorem MulLECancellable.isLeftRegular {R : Type u_1} [Mul R] [PartialOrder R] {a : R} (ha : MulLECancellable a) : IsLeftRegular a

theorem AddLECancellable.isAddLeftRegular {R : Type u_1} [Add R] [PartialOrder R] {a : R} (ha : AddLECancellable a) : IsAddLeftRegular a

theorem IsLeftRegular.right_of_commute {R : Type u_1} [Mul R] {a : R} (ca : ∀ (b : R), Commute a b) (h : IsLeftRegular a) : IsRightRegular a

theorem IsRightRegular.left_of_commute {R : Type u_1} [Mul R] {a : R} (ca : ∀ (b : R), Commute a b) (h : IsRightRegular a) : IsLeftRegular a

theorem Commute.isRightRegular_iff {R : Type u_1} [Mul R] {a : R} (ca : ∀ (b : R), Commute a b) : IsRightRegular a ↔ IsLeftRegular a

theorem Commute.isRegular_iff {R : Type u_1} [Mul R] {a : R} (ca : ∀ (b : R), Commute a b) : IsRegular a ↔ IsLeftRegular a

theorem IsLeftRegular.mul {R : Type u_1} [Semigroup R] {a b : R} (lra : IsLeftRegular a) (lrb : IsLeftRegular b) : IsLeftRegular (a * b)

In a semigroup, the product of left-regular elements is left-regular.

theorem IsAddLeftRegular.add {R : Type u_1} [AddSemigroup R] {a b : R} (lra : IsAddLeftRegular a) (lrb : IsAddLeftRegular b) : IsAddLeftRegular (a + b)

In an additive semigroup, the sum of add-left-regular elements is add-left.regular.

theorem IsRightRegular.mul {R : Type u_1} [Semigroup R] {a b : R} (rra : IsRightRegular a) (rrb : IsRightRegular b) : IsRightRegular (a * b)

In a semigroup, the product of right-regular elements is right-regular.

theorem IsAddRightRegular.add {R : Type u_1} [AddSemigroup R] {a b : R} (rra : IsAddRightRegular a) (rrb : IsAddRightRegular b) : IsAddRightRegular (a + b)

In an additive semigroup, the sum of add-right-regular elements is add-right-regular.

theorem IsRegular.mul {R : Type u_1} [Semigroup R] {a b : R} (rra : IsRegular a) (rrb : IsRegular b) : IsRegular (a * b)

In a semigroup, the product of regular elements is regular.

theorem IsAddRegular.add {R : Type u_1} [AddSemigroup R] {a b : R} (rra : IsAddRegular a) (rrb : IsAddRegular b) : IsAddRegular (a + b)

In an additive semigroup, the sum of add-regular elements is add-regular.

theorem IsLeftRegular.of_mul {R : Type u_1} [Semigroup R] {a b : R} (ab : IsLeftRegular (a * b)) : IsLeftRegular b

If an element b becomes left-regular after multiplying it on the left by a left-regular element, then b is left-regular.

theorem IsAddLeftRegular.of_add {R : Type u_1} [AddSemigroup R] {a b : R} (ab : IsAddLeftRegular (a + b)) : IsAddLeftRegular b

If an element b becomes add-left-regular after adding to it on the left an add-left-regular element, then b is add-left-regular.

@[simp]

theorem mul_isLeftRegular_iff {R : Type u_1} [Semigroup R] {a : R} (b : R) (ha : IsLeftRegular a) : IsLeftRegular (a * b) ↔ IsLeftRegular b

An element is left-regular if and only if multiplying it on the left by a left-regular element is left-regular.

@[simp]

theorem add_isAddLeftRegular_iff {R : Type u_1} [AddSemigroup R] {a : R} (b : R) (ha : IsAddLeftRegular a) : IsAddLeftRegular (a + b) ↔ IsAddLeftRegular b

An element is add-left-regular if and only if adding to it on the left an add-left-regular element is add-left-regular.

theorem IsRightRegular.of_mul {R : Type u_1} [Semigroup R] {a b : R} (ab : IsRightRegular (b * a)) : IsRightRegular b

If an element b becomes right-regular after multiplying it on the right by a right-regular element, then b is right-regular.

theorem IsAddRightRegular.of_add {R : Type u_1} [AddSemigroup R] {a b : R} (ab : IsAddRightRegular (b + a)) : IsAddRightRegular b

If an element b becomes add-right-regular after adding to it on the right an add-right-regular element, then b is add-right-regular.

@[simp]

theorem mul_isRightRegular_iff {R : Type u_1} [Semigroup R] {a : R} (b : R) (ha : IsRightRegular a) : IsRightRegular (b * a) ↔ IsRightRegular b

An element is right-regular if and only if multiplying it on the right with a right-regular element is right-regular.

@[simp]

theorem add_isAddRightRegular_iff {R : Type u_1} [AddSemigroup R] {a : R} (b : R) (ha : IsAddRightRegular a) : IsAddRightRegular (b + a) ↔ IsAddRightRegular b

An element is add-right-regular if and only if adding it on the right to an add-right-regular element is add-right-regular.

theorem isRegular_mul_and_mul_iff {R : Type u_1} [Semigroup R] {a b : R} : IsRegular (a * b) ∧ IsRegular (b * a) ↔ IsRegular a ∧ IsRegular b

Two elements a and b are regular if and only if both products a * b and b * a are regular.

theorem isAddRegular_add_and_add_iff {R : Type u_1} [AddSemigroup R] {a b : R} : IsAddRegular (a + b) ∧ IsAddRegular (b + a) ↔ IsAddRegular a ∧ IsAddRegular b

Two elements a and b are add-regular if and only if both sums a + b and b + a are add-regular.

theorem IsRegular.and_of_mul_of_mul {R : Type u_1} [Semigroup R] {a b : R} (ab : IsRegular (a * b)) (ba : IsRegular (b * a)) : IsRegular a ∧ IsRegular b

The "most used" implication of mul_and_mul_iff, with split hypotheses, instead of ∧.

theorem IsAddRegular.and_of_add_of_add {R : Type u_1} [AddSemigroup R] {a b : R} (ab : IsAddRegular (a + b)) (ba : IsAddRegular (b + a)) : IsAddRegular a ∧ IsAddRegular b

The "most used" implication of add_and_add_iff, with split hypotheses, instead of ∧.

theorem IsLeftRegular.subsingleton {R : Type u_1} [MulZeroClass R] (h : IsLeftRegular 0) : Subsingleton R

The element 0 is left-regular if and only if R is trivial.

theorem IsRightRegular.subsingleton {R : Type u_1} [MulZeroClass R] (h : IsRightRegular 0) : Subsingleton R

The element 0 is right-regular if and only if R is trivial.

theorem IsRegular.subsingleton {R : Type u_1} [MulZeroClass R] (h : IsRegular 0) : Subsingleton R

The element 0 is regular if and only if R is trivial.

theorem isLeftRegular_zero_iff_subsingleton {R : Type u_1} [MulZeroClass R] : IsLeftRegular 0 ↔ Subsingleton R

The element 0 is left-regular if and only if R is trivial.

theorem not_isLeftRegular_zero_iff {R : Type u_1} [MulZeroClass R] : ¬IsLeftRegular 0 ↔ Nontrivial R

In a non-trivial MulZeroClass, the 0 element is not left-regular.

theorem isRightRegular_zero_iff_subsingleton {R : Type u_1} [MulZeroClass R] : IsRightRegular 0 ↔ Subsingleton R

The element 0 is right-regular if and only if R is trivial.

theorem not_isRightRegular_zero_iff {R : Type u_1} [MulZeroClass R] : ¬IsRightRegular 0 ↔ Nontrivial R

In a non-trivial MulZeroClass, the 0 element is not right-regular.

theorem isRegular_iff_subsingleton {R : Type u_1} [MulZeroClass R] : IsRegular 0 ↔ Subsingleton R

The element 0 is regular if and only if R is trivial.

theorem IsLeftRegular.ne_zero {R : Type u_1} [MulZeroClass R] {a : R} [Nontrivial R] (la : IsLeftRegular a) : a ≠ 0

A left-regular element of a Nontrivial MulZeroClass is non-zero.

theorem IsRightRegular.ne_zero {R : Type u_1} [MulZeroClass R] {a : R} [Nontrivial R] (ra : IsRightRegular a) : a ≠ 0

A right-regular element of a Nontrivial MulZeroClass is non-zero.

theorem IsRegular.ne_zero {R : Type u_1} [MulZeroClass R] {a : R} [Nontrivial R] (la : IsRegular a) : a ≠ 0

A regular element of a Nontrivial MulZeroClass is non-zero.

theorem not_isLeftRegular_zero {R : Type u_1} [MulZeroClass R] [nR : Nontrivial R] : ¬IsLeftRegular 0

In a non-trivial ring, the element 0 is not left-regular -- with typeclasses.

theorem not_isRightRegular_zero {R : Type u_1} [MulZeroClass R] [nR : Nontrivial R] : ¬IsRightRegular 0

In a non-trivial ring, the element 0 is not right-regular -- with typeclasses.

theorem not_isRegular_zero {R : Type u_1} [MulZeroClass R] [Nontrivial R] : ¬IsRegular 0

In a non-trivial ring, the element 0 is not regular -- with typeclasses.

@[simp]

theorem IsLeftRegular.mul_left_eq_zero_iff {R : Type u_1} [MulZeroClass R] {a b : R} (hb : IsLeftRegular b) : b * a = 0 ↔ a = 0

@[simp]

theorem IsRightRegular.mul_right_eq_zero_iff {R : Type u_1} [MulZeroClass R] {a b : R} (hb : IsRightRegular b) : a * b = 0 ↔ a = 0

theorem isRegular_one {R : Type u_1} [MulOneClass R] : IsRegular 1

If multiplying by 1 on either side is the identity, 1 is regular.

theorem isAddRegular_zero {R : Type u_1} [AddZeroClass R] : IsAddRegular 0

If adding 0 on either side is the identity, 0 is regular.

theorem isRegular_mul_iff {R : Type u_1} [CommSemigroup R] {a b : R} : IsRegular (a * b) ↔ IsRegular a ∧ IsRegular b

A product is regular if and only if the factors are.

theorem isAddRegular_add_iff {R : Type u_1} [AddCommSemigroup R] {a b : R} : IsAddRegular (a + b) ↔ IsAddRegular a ∧ IsAddRegular b

A sum is add-regular if and only if the summands are.

theorem isLeftRegular_of_mul_eq_one {R : Type u_1} [Monoid R] {a b : R} (h : b * a = 1) : IsLeftRegular a

An element admitting a left inverse is left-regular.

theorem isAddLeftRegular_of_add_eq_zero {R : Type u_1} [AddMonoid R] {a b : R} (h : b + a = 0) : IsAddLeftRegular a

An element admitting a left additive opposite is add-left-regular.

theorem isRightRegular_of_mul_eq_one {R : Type u_1} [Monoid R] {a b : R} (h : a * b = 1) : IsRightRegular a

An element admitting a right inverse is right-regular.

theorem isAddRightRegular_of_add_eq_zero {R : Type u_1} [AddMonoid R] {a b : R} (h : a + b = 0) : IsAddRightRegular a

An element admitting a right additive opposite is add-right-regular.

theorem Units.isRegular {R : Type u_1} [Monoid R] (a : Rˣ) : IsRegular ↑a

If R is a monoid, an element in Rˣ is regular.

theorem AddUnits.isAddRegular {R : Type u_1} [AddMonoid R] (a : AddUnits R) : IsAddRegular ↑a

If R is an additive monoid, an element in add_units R is add-regular.

theorem IsUnit.isRegular {R : Type u_1} [Monoid R] {a : R} (ua : IsUnit a) : IsRegular a

A unit in a monoid is regular.

theorem IsAddUnit.isAddRegular {R : Type u_1} [AddMonoid R] {a : R} (ua : IsAddUnit a) : IsAddRegular a

An additive unit in an additive monoid is add-regular.

theorem IsLeftRegular.pow {R : Type u_1} [Monoid R] {a : R} (n : ℕ) (rla : IsLeftRegular a) : IsLeftRegular (a ^ n)

Any power of a left-regular element is left-regular.

theorem IsRightRegular.pow {R : Type u_1} [Monoid R] {a : R} (n : ℕ) (rra : IsRightRegular a) : IsRightRegular (a ^ n)

Any power of a right-regular element is right-regular.

theorem IsRegular.pow {R : Type u_1} [Monoid R] {a : R} (n : ℕ) (ra : IsRegular a) : IsRegular (a ^ n)

Any power of a regular element is regular.

theorem IsLeftRegular.pow_iff {R : Type u_1} [Monoid R] {a : R} {n : ℕ} (n0 : 0 < n) : IsLeftRegular (a ^ n) ↔ IsLeftRegular a

An element a is left-regular if and only if a positive power of a is left-regular.

theorem IsRightRegular.pow_iff {R : Type u_1} [Monoid R] {a : R} {n : ℕ} (n0 : 0 < n) : IsRightRegular (a ^ n) ↔ IsRightRegular a

An element a is right-regular if and only if a positive power of a is right-regular.

theorem IsRegular.pow_iff {R : Type u_1} [Monoid R] {a : R} {n : ℕ} (n0 : 0 < n) : IsRegular (a ^ n) ↔ IsRegular a

An element a is regular if and only if a positive power of a is regular.

theorem IsLeftRegular.all {R : Type u_1} [Mul R] [IsLeftCancelMul R] (g : R) : IsLeftRegular g

If all multiplications cancel on the left then every element is left-regular.

theorem IsAddLeftRegular.all {R : Type u_1} [Add R] [IsLeftCancelAdd R] (g : R) : IsAddLeftRegular g

If all additions cancel on the left then every element is add-left-regular.

theorem IsRightRegular.all {R : Type u_1} [Mul R] [IsRightCancelMul R] (g : R) : IsRightRegular g

If all multiplications cancel on the right then every element is right-regular.

theorem IsAddRightRegular.all {R : Type u_1} [Add R] [IsRightCancelAdd R] (g : R) : IsAddRightRegular g

If all additions cancel on the right then every element is add-right-regular.

theorem IsRegular.all {R : Type u_1} [Mul R] [IsCancelMul R] (g : R) : IsRegular g

If all multiplications cancel then every element is regular.

theorem IsAddRegular.all {R : Type u_1} [Add R] [IsCancelAdd R] (g : R) : IsAddRegular g

If all additions cancel then every element is add-regular.

theorem isRegular_of_ne_zero {R : Type u_1} [MulZeroClass R] [IsCancelMulZero R] {a : R} (a0 : a ≠ 0) : IsRegular a

Non-zero elements of an integral domain are regular.

theorem isRegular_iff_ne_zero {R : Type u_1} [MulZeroClass R] [IsCancelMulZero R] {a : R} [Nontrivial R] : IsRegular a ↔ a ≠ 0

In a non-trivial integral domain, an element is regular iff it is non-zero.