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 IsSMulRegular.

There are very limited typeclass assumptions on R and M, but the "mathematical" case of interest is a commutative ring R acting on 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 IsLeftRegular defined in Algebra/Regular. Lemma isLeftRegular_iff shows that indeed the two notions coincide.

def IsSMulRegular {R : Type u_1} (M : Type u_3) [SMul 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

• IsSMulRegular M c = Function.Injective fun (x : M) => c • x

theorem IsLeftRegular.isSMulRegular {R : Type u_1} [Mul R] {c : R} (h : IsLeftRegular c) : IsSMulRegular R c

theorem isLeftRegular_iff {R : Type u_1} [Mul R] {a : R} : IsLeftRegular a ↔ IsSMulRegular R a

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

theorem IsRightRegular.isSMulRegular {R : Type u_1} [Mul R] {c : R} (h : IsRightRegular c) : IsSMulRegular R (MulOpposite.op c)

theorem isRightRegular_iff {R : Type u_1} [Mul R] {a : R} : IsRightRegular a ↔ IsSMulRegular R (MulOpposite.op a)

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

theorem IsSMulRegular.smul {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (ra : IsSMulRegular M a) (rs : IsSMulRegular M s) : IsSMulRegular M (a • s)

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

theorem IsSMulRegular.of_smul {R : Type u_1} {S : Type u_2} {M : Type u_3} {s : S} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M 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 IsSMulRegular.smul_iff {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (b : S) (ha : IsSMulRegular M a) : IsSMulRegular M (a • b) ↔ IsSMulRegular M b

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

theorem IsSMulRegular.isLeftRegular {R : Type u_1} [Mul R] {a : R} (h : IsSMulRegular R a) : IsLeftRegular a

theorem IsSMulRegular.isRightRegular {R : Type u_1} [Mul R] {a : R} (h : IsSMulRegular R (MulOpposite.op a)) : IsRightRegular a

theorem IsSMulRegular.mul {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ra : IsSMulRegular M a) (rb : IsSMulRegular M b) : IsSMulRegular M (a * b)

theorem IsSMulRegular.of_mul {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) : IsSMulRegular M b

@[simp]

theorem IsSMulRegular.mul_iff_right {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ha : IsSMulRegular M a) : IsSMulRegular M (a * b) ↔ IsSMulRegular M b

theorem IsSMulRegular.mul_and_mul_iff {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] : IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b

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

theorem IsSMulRegular.of_injective {R : Type u_1} {M : Type u_3} [SMul R M] {N : Type u_4} {F : Type u_5} [SMul R N] [FunLike F M N] [MulActionHomClass F R M N] (f : F) {r : R} (h1 : Function.Injective ⇑f) (h2 : IsSMulRegular N r) : IsSMulRegular M r

@[simp]

theorem IsSMulRegular.one {R : Type u_1} (M : Type u_3) [Monoid R] [MulAction R M] : IsSMulRegular M 1

One is always M-regular.

theorem IsSMulRegular.of_mul_eq_one {R : Type u_1} {M : Type u_3} {a b : R} [Monoid R] [MulAction R M] (h : a * b = 1) : IsSMulRegular M b

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

theorem IsSMulRegular.pow {R : Type u_1} {M : Type u_3} {a : R} [Monoid R] [MulAction R M] (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n)

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

theorem IsSMulRegular.pow_iff {R : Type u_1} {M : Type u_3} {a : R} [Monoid R] [MulAction R M] {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a

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

theorem IsSMulRegular.of_smul_eq_one {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [Monoid S] [SMul R M] [SMul R S] [MulAction S M] [IsScalarTower R S M] (h : a • s = 1) : IsSMulRegular M s

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

theorem IsSMulRegular.subsingleton {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (h : IsSMulRegular M 0) : Subsingleton M

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

theorem IsSMulRegular.zero_iff_subsingleton {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] : IsSMulRegular M 0 ↔ Subsingleton M

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

theorem IsSMulRegular.not_zero_iff {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] : ¬IsSMulRegular M 0 ↔ Nontrivial M

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

theorem IsSMulRegular.zero {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [sM : Subsingleton M] : IsSMulRegular M 0

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

theorem IsSMulRegular.not_zero {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [nM : Nontrivial M] : ¬IsSMulRegular M 0

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

theorem IsSMulRegular.mul_iff {R : Type u_1} {M : Type u_3} {a b : R} [CommSemigroup R] [SMul R M] [IsScalarTower R R M] : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b

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

theorem isSMulRegular_of_group {R : Type u_1} {G : Type u_4} [Group G] [MulAction G R] (g : G) : IsSMulRegular R 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 LeftCancelSMul typeclass.

theorem Units.isSMulRegular {R : Type u_1} (M : Type u_3) [Monoid R] [MulAction R M] (a : Rˣ) : IsSMulRegular M ↑a

Any element in Rˣ is M-regular.

theorem IsUnit.isSMulRegular {R : Type u_1} (M : Type u_3) {a : R} [Monoid R] [MulAction R M] (ua : IsUnit a) : IsSMulRegular M a

A unit is M-regular.

theorem IsSMulRegular.eq_zero_of_smul_eq_zero {R : Type u_1} {M : Type u_3} [Zero M] [SMulZeroClass R M] {r : R} {x : M} (h1 : IsSMulRegular M r) (h2 : r • x = 0) : x = 0

theorem Equiv.isSMulRegular_congr {R : Type u_4} {S : Type u_5} {M : Type u_6} {M' : Type u_7} [SMul R M] [SMul S M'] {e : M ≃ M'} {r : R} {s : S} (h : ∀ (x : M), e (r • x) = s • e x) : IsSMulRegular M r ↔ IsSMulRegular M' s