Submodules of a module

In this file we define

• Submodule R M : a subset of a Module M that contains zero and is closed with respect to addition and scalar multiplication.

• Subspace k M : an abbreviation for Submodule assuming that k is a Field.

Tags

submodule, subspace, linear map

structure Submodule (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] extends AddSubmonoid M, SubMulAction R M : Type v

A submodule of a module is one which is closed under vector operations. This is a sufficient condition for the subset of vectors in the submodule to themselves form a module.

• carrier : Set M
• add_mem' {a b : M} : a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
• zero_mem' : 0 ∈ self.carrier
• smul_mem' (c : R) {x : M} : x ∈ self.carrier → c • x ∈ self.carrier

instance Submodule.setLike {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : SetLike (Submodule R M) M

Equations

• Submodule.setLike = { coe := fun (s : Submodule R M) => s.carrier, coe_injective' := ⋯ }

@[simp]

theorem Submodule.carrier_eq_coe {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (s : Submodule R M) : s.carrier = ↑s

def Submodule.ofClass {S : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) : Submodule R M

The actual Submodule obtained from an element of a SMulMemClass and AddSubmonoidClass.

Equations

• Submodule.ofClass s = { carrier := ↑s, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

@[simp]

theorem Submodule.coe_ofClass {S : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) : ↑(ofClass s) = ↑s

def Submodule.ofLinearComb {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (C : Set M) (nonempty : C.Nonempty) (linearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) : Submodule R M

Construct a submodule from closure under two-element linear combinations. I.e., a nonempty set closed under two-element linear combinations is a submodule.

Equations

• Submodule.ofLinearComb C nonempty linearComb = { carrier := C, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

@[simp]

theorem Submodule.coe_ofLinearComb {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (C : Set M) (nonempty : C.Nonempty) (linearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C) : ↑(ofLinearComb C nonempty linearComb) = C

@[instance 100]

instance Submodule.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallHSMul {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : CanLift (Set M) (Submodule R M) SetLike.coe fun (s : Set M) => 0 ∈ s ∧ (∀ {x y : M}, x ∈ s → y ∈ s → x + y ∈ s) ∧ ∀ (r : R) {x : M}, x ∈ s → r • x ∈ s

instance Submodule.addSubmonoidClass {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : AddSubmonoidClass (Submodule R M) M

instance Submodule.smulMemClass {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : SMulMemClass (Submodule R M) R M

@[simp]

theorem Submodule.mem_toAddSubmonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (x : M) : x ∈ p.toAddSubmonoid ↔ x ∈ p

@[simp]

theorem Submodule.mem_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {S : AddSubmonoid M} {x : M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier) : x ∈ { toAddSubmonoid := S, smul_mem' := h } ↔ x ∈ S

@[simp]

theorem Submodule.coe_set_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (S : AddSubmonoid M) (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier) : ↑{ toAddSubmonoid := S, smul_mem' := h } = ↑S

@[simp]

theorem Submodule.eta {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} (h : ∀ (c : R) {x : M}, x ∈ p.carrier → c • x ∈ p.carrier) : { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := h } = p

@[simp]

theorem Submodule.mk_le_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {S S' : AddSubmonoid M} (h : ∀ (c : R) {x : M}, x ∈ S.carrier → c • x ∈ S.carrier) (h' : ∀ (c : R) {x : M}, x ∈ S'.carrier → c • x ∈ S'.carrier) : { toAddSubmonoid := S, smul_mem' := h } ≤ { toAddSubmonoid := S', smul_mem' := h' } ↔ S ≤ S'

theorem Submodule.ext {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (h : ∀ (x : M), x ∈ p ↔ x ∈ q) : p = q

theorem Submodule.ext_iff {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p = q ↔ ∀ (x : M), x ∈ p ↔ x ∈ q

@[deprecated SetLike.coe_set_eq (since := "2025-04-20")]

theorem Submodule.carrier_inj {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p.carrier = q.carrier ↔ p = q

def Submodule.copy {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (s : Set M) (hs : s = ↑p) : Submodule R M

Copy of a submodule with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

• p.copy s hs = { carrier := s, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

@[simp]

theorem Submodule.coe_copy {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (s : Set M) (hs : s = ↑p) : ↑(p.copy s hs) = s

theorem Submodule.copy_eq {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (S : Submodule R M) (s : Set M) (hs : s = ↑S) : S.copy s hs = S

theorem Submodule.toAddSubmonoid_injective {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : Function.Injective toAddSubmonoid

@[simp]

theorem Submodule.toAddSubmonoid_inj {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p.toAddSubmonoid = q.toAddSubmonoid ↔ p = q

@[simp]

theorem Submodule.coe_toAddSubmonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ↑p.toAddSubmonoid = ↑p

theorem Submodule.toSubMulAction_injective {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : Function.Injective toSubMulAction

theorem Submodule.toSubMulAction_inj {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p.toSubMulAction = q.toSubMulAction ↔ p = q

@[simp]

theorem Submodule.coe_toSubMulAction {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ↑p.toSubMulAction = ↑p

@[instance 75]

instance SMulMemClass.toModule {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {A : Type u_1} [SetLike A M] [AddSubmonoidClass A M] [SMulMemClass A R M] (S' : A) : Module R ↥S'

A submodule of a Module is a Module.

Equations

• SMulMemClass.toModule S' = { toMulAction := SubMulAction.SMulMemClass.toMulAction S', smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }

def SMulMemClass.toModule' (S : Type u_2) (R' : Type u_3) (R : Type u_4) (A : Type u_5) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] [SetLike S A] [AddSubmonoidClass S A] [SMulMemClass S R A] (s : S) : Module R' ↥s

This can't be an instance because Lean wouldn't know how to find R, but we can still use this to manually derive Module on specific types.

Equations

• SMulMemClass.toModule' S R' R A s = SMulMemClass.toModule s

theorem Submodule.mem_carrier {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} : x ∈ p.carrier ↔ x ∈ ↑p

theorem Submodule.zero_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : 0 ∈ p

theorem Submodule.add_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x y : M} (h₁ : x ∈ p) (h₂ : y ∈ p) : x + y ∈ p

theorem Submodule.smul_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (r : R) (h : x ∈ p) : r • x ∈ p

theorem Submodule.smul_of_tower_mem {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (h : x ∈ p) : r • x ∈ p

@[simp]

theorem Submodule.smul_mem_iff' {G : Type u''} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [Group G] [MulAction G M] [SMul G R] [IsScalarTower G R M] (g : G) : g • x ∈ p ↔ x ∈ p

@[simp]

theorem Submodule.smul_mem_iff'' {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {r : R} {x : M} [Invertible r] : r • x ∈ p ↔ x ∈ p

theorem Submodule.smul_mem_iff_of_isUnit {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {r : R} {x : M} (hr : IsUnit r) : r • x ∈ p ↔ x ∈ p

instance Submodule.add {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Add ↥p

Equations

• p.add = { add := fun (x y : ↥p) => ⟨↑x + ↑y, ⋯⟩ }

instance Submodule.zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Zero ↥p

Equations

• p.zero = { zero := ⟨0, ⋯⟩ }

instance Submodule.inhabited {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Inhabited ↥p

Equations

• p.inhabited = { default := 0 }

instance Submodule.smul {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] : SMul S ↥p

Equations

• p.smul = { smul := fun (c : S) (x : ↥p) => ⟨c • ↑x, ⋯⟩ }

instance Submodule.isScalarTower {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] : IsScalarTower S R ↥p

instance Submodule.isScalarTower' {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {S' : Type u_1} [SMul S R] [SMul S M] [SMul S' R] [SMul S' M] [SMul S S'] [IsScalarTower S' R M] [IsScalarTower S S' M] [IsScalarTower S R M] : IsScalarTower S S' ↥p

theorem Submodule.nonempty {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : (↑p).Nonempty

@[simp]

theorem Submodule.mk_eq_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} (h : x ∈ p) : ⟨x, h⟩ = 0 ↔ x = 0

theorem Submodule.coe_eq_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} {x : ↥p} : ↑x = 0 ↔ x = 0

@[simp]

theorem Submodule.coe_add {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x y : ↥p) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem Submodule.coe_zero {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} : ↑0 = 0

theorem Submodule.coe_smul {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (r : R) (x : ↥p) : ↑(r • x) = r • ↑x

@[simp]

theorem Submodule.coe_smul_of_tower {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (x : ↥p) : ↑(r • x) = r • ↑x

theorem Submodule.coe_mk {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : M) (hx : x ∈ p) : ↑⟨x, hx⟩ = x

theorem Submodule.coe_mem {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : ↥p) : ↑x ∈ p

instance Submodule.addCommMonoid {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : AddCommMonoid ↥p

Equations

• p.addCommMonoid = p.toAddCommMonoid

instance Submodule.module' {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [Semiring S] [SMul S R] [Module S M] [IsScalarTower S R M] : Module S ↥p

Equations

• p.module' = { toSMul := p.toSubMulAction.smul', one_smul := ⋯, mul_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }

instance Submodule.module {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Module R ↥p

Equations

• p.module = p.module'

instance Submodule.addSubgroupClass {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] : AddSubgroupClass (Submodule R M) M

theorem Submodule.neg_mem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} (hx : x ∈ p) : -x ∈ p

def Submodule.toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : AddSubgroup M

Reinterpret a submodule as an additive subgroup.

Equations

• p.toAddSubgroup = { toAddSubmonoid := p.toAddSubmonoid, neg_mem' := ⋯ }

@[simp]

theorem Submodule.coe_toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : ↑p.toAddSubgroup = ↑p

@[simp]

theorem Submodule.mem_toAddSubgroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} : x ∈ p.toAddSubgroup ↔ x ∈ p

theorem Submodule.toAddSubgroup_injective {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} : Function.Injective toAddSubgroup

@[simp]

theorem Submodule.toAddSubgroup_inj {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p p' : Submodule R M) : p.toAddSubgroup = p'.toAddSubgroup ↔ p = p'

theorem Submodule.sub_mem {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x y : M} : x ∈ p → y ∈ p → x - y ∈ p

theorem Submodule.neg_mem_iff {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x : M} : -x ∈ p ↔ x ∈ p

theorem Submodule.add_mem_iff_left {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x y : M} : y ∈ p → (x + y ∈ p ↔ x ∈ p)

theorem Submodule.add_mem_iff_right {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x y : M} : x ∈ p → (x + y ∈ p ↔ y ∈ p)

theorem Submodule.coe_neg {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x : ↥p) : ↑(-x) = -↑x

theorem Submodule.coe_sub {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) (x y : ↥p) : ↑(x - y) = ↑x - ↑y

theorem Submodule.sub_mem_iff_left {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x y : M} (hy : y ∈ p) : x - y ∈ p ↔ x ∈ p

theorem Submodule.sub_mem_iff_right {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) {x y : M} (hx : x ∈ p) : x - y ∈ p ↔ y ∈ p

instance Submodule.addCommGroup {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : AddCommGroup ↥p

Equations

• p.addCommGroup = p.toAddSubgroup.toAddCommGroup

@[instance 75]

instance SubmoduleClass.module' {S : Type u'} {R : Type u} {M : Type v} {T : Type u_1} [Semiring R] [AddCommMonoid M] [Semiring S] [Module R M] [SMul S R] [Module S M] [IsScalarTower S R M] [SetLike T M] [AddSubmonoidClass T M] [SMulMemClass T R M] (t : T) : Module S ↥t

Equations

• SubmoduleClass.module' t = { toSMul := SetLike.smul' t, one_smul := ⋯, mul_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }

@[instance 75]

instance SubmoduleClass.module {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) : Module R ↥s

Equations

• SubmoduleClass.module s = SubmoduleClass.module' s