mathlib documentation

algebra.regular.smul

Action of regular elements on a module #

We introduce M-regular elements, in the context of an R-module M. The corresponding predicate is called is_smul_regular.

There are very limited typeclass assumptions on R and M, but the "mathematical" case of interest is a commutative ring R acting an a module M. Since the properties are "multiplicative", there is no actual requirement of having an addition, but there is a zero in both R and M. Smultiplications involving 0 are, of course, all trivial.

The defining property is that an element a ∈ R is M-regular if the smultiplication map M → M, defined by m ↦ a • m, is injective.

This property is the direct generalization to modules of the property is_left_regular defined in algebra/regular. Lemma is_smul_regular.is_left_regular_iff shows that indeed the two notions coincide.

def is_smul_regular {R : Type u_1} (M : Type u_3) [has_scalar R M] (c : R) :
Prop

An M-regular element is an element c such that multiplication on the left by c is an injective map M → M.

Equations
theorem is_left_regular.is_smul_regular {R : Type u_1} [has_mul R] {c : R} (h : is_left_regular c) :
theorem is_left_regular_iff {R : Type u_1} [has_mul R] {a : R} :

Left-regular multiplication on R is equivalent to R-regularity of R itself.

theorem is_right_regular.is_smul_regular {R : Type u_1} [has_mul R] {c : R} (h : is_right_regular c) :
theorem is_right_regular_iff {R : Type u_1} [has_mul R] {a : R} :

Right-regular multiplication on R is equivalent to Rᵒᵖ-regularity of R itself.

theorem is_smul_regular.smul {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [has_scalar R M] [has_scalar R S] [has_scalar S M] [is_scalar_tower R S M] (ra : is_smul_regular M a) (rs : is_smul_regular M s) :

The product of M-regular elements is M-regular.

theorem is_smul_regular.of_smul {R : Type u_1} {S : Type u_2} {M : Type u_3} {s : S} [has_scalar R M] [has_scalar R S] [has_scalar S M] [is_scalar_tower R S M] (a : R) (ab : is_smul_regular M (a s)) :

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

@[simp]
theorem is_smul_regular.smul_iff {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} [has_scalar R M] [has_scalar R S] [has_scalar S M] [is_scalar_tower R S M] (b : S) (ha : is_smul_regular M a) :

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

theorem is_smul_regular.is_left_regular {R : Type u_1} [has_mul R] {a : R} (h : is_smul_regular R a) :
theorem is_smul_regular.is_right_regular {R : Type u_1} [has_mul R] {a : R} (h : is_smul_regular R (opposite.op a)) :
@[simp]
theorem is_smul_regular.one {R : Type u_1} (M : Type u_3) [monoid R] [mul_action R M] :

One is M-regular always.

theorem is_smul_regular.mul {R : Type u_1} {M : Type u_3} {a b : R} [monoid R] [mul_action R M] (ra : is_smul_regular M a) (rb : is_smul_regular M b) :
theorem is_smul_regular.of_mul {R : Type u_1} {M : Type u_3} {a b : R} [monoid R] [mul_action R M] (ab : is_smul_regular M (a * b)) :
@[simp]
theorem is_smul_regular.mul_iff_right {R : Type u_1} {M : Type u_3} {a b : R} [monoid R] [mul_action R M] (ha : is_smul_regular M a) :
theorem is_smul_regular.mul_and_mul_iff {R : Type u_1} {M : Type u_3} {a b : R} [monoid R] [mul_action R M] :

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

theorem is_smul_regular.pow {R : Type u_1} {M : Type u_3} {a : R} [monoid R] [mul_action R M] (n : ) (ra : is_smul_regular M a) :

Any power of an M-regular element is M-regular.

theorem is_smul_regular.pow_iff {R : Type u_1} {M : Type u_3} {a : R} [monoid R] [mul_action R M] {n : } (n0 : 0 < n) :

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

theorem is_smul_regular.subsingleton {R : Type u_1} {M : Type u_3} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] (h : is_smul_regular M 0) :

The element 0 is M-regular if and only if M is trivial.

The element 0 is M-regular if and only if M is trivial.

theorem is_smul_regular.not_zero_iff {R : Type u_1} {M : Type u_3} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] :

The 0 element is not M-regular, on a non-trivial module.

theorem is_smul_regular.zero {R : Type u_1} {M : Type u_3} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] [sM : subsingleton M] :

The element 0 is M-regular when M is trivial.

theorem is_smul_regular.not_zero {R : Type u_1} {M : Type u_3} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] [nM : nontrivial M] :

The 0 element is not M-regular, on a non-trivial module.

theorem is_smul_regular.of_smul_eq_one {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [monoid_with_zero R] [monoid_with_zero S] [has_zero M] [mul_action_with_zero R M] [mul_action_with_zero R S] [mul_action_with_zero S M] [is_scalar_tower R S M] (h : a s = 1) :

An element of S admitting a left inverse in R is M-regular.

theorem is_smul_regular.of_mul_eq_one {R : Type u_1} {M : Type u_3} {a b : R} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] (h : a * b = 1) :

An element of R admitting a left inverse is M-regular.

theorem is_smul_regular.mul_iff {R : Type u_1} {M : Type u_3} {a b : R} [comm_monoid R] [mul_action R M] :

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

theorem is_smul_regular_of_group {R : Type u_1} {G : Type u_4} [group G] [mul_action G R] (g : G) :

An element of a group acting on a Type is regular. This relies on the availability of the inverse given by groups, since there is no left_cancel_smul typeclass.

theorem units.is_smul_regular {R : Type u_1} (M : Type u_3) [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] (a : units R) :

Any element in units R is M-regular.

theorem is_unit.is_smul_regular {R : Type u_1} (M : Type u_3) {a : R} [monoid_with_zero R] [has_zero M] [mul_action_with_zero R M] (ua : is_unit a) :

A unit is M-regular.