# Documentation

Mathlib.Algebra.Module.Basic

# Modules over a ring #

In this file we define

• Module R M : an additive commutative monoid M is a Module over a Semiring R if for r : R and x : M their "scalar multiplication" r • x : M is defined, and the operation • satisfies some natural associativity and distributivity axioms similar to those on a ring.

## Implementation notes #

In typical mathematical usage, our definition of Module corresponds to "semimodule", and the word "module" is reserved for Module R M where R is a Ring and M an AddCommGroup. If R is a Field and M an AddCommGroup, M would be called an R-vector space. Since those assumptions can be made by changing the typeclasses applied to R and M, without changing the axioms in Module, mathlib calls everything a Module.

In older versions of mathlib3, we had separate semimodule and vector_space abbreviations. This caused inference issues in some cases, while not providing any real advantages, so we decided to use a canonical Module typeclass throughout.

## Tags #

semimodule, module, vector space

theorem Module.ext {R : Type u} {M : Type v} :
∀ {inst : } {inst_1 : } (x y : Module R M), SMul.smul = SMul.smulx = y
theorem Module.ext_iff {R : Type u} {M : Type v} :
∀ {inst : } {inst_1 : } (x y : Module R M), x = y SMul.smul = SMul.smul
class Module (R : Type u) (M : Type v) [] [] extends :
Type (max u v)
• smul : RMM
• one_smul : ∀ (b : M), 1 b = b
• mul_smul : ∀ (x y : R) (b : M), (x * y) b = x y b
• smul_zero : ∀ (a : R), a 0 = 0
• smul_add : ∀ (a : R) (x y : M), a (x + y) = a x + a y
• add_smul : ∀ (r s_1 : R) (x : M), (r + s_1) x = r x + s_1 x

Scalar multiplication distributes over addition from the right.

• zero_smul : ∀ (x : M), 0 x = 0

Scalar multiplication by zero gives zero.

A module is a generalization of vector spaces to a scalar semiring. It consists of a scalar semiring R and an additive monoid of "vectors" M, connected by a "scalar multiplication" operation r • x : M (where r : R and x : M) with some natural associativity and distributivity axioms similar to those on a ring.

Instances
instance Module.toMulActionWithZero {R : Type u_2} {M : Type u_5} [] [] [Module R M] :

A module over a semiring automatically inherits a MulActionWithZero structure.

instance AddCommMonoid.natModule {M : Type u_5} [] :
theorem AddMonoid.End.nat_cast_def {M : Type u_5} [] (n : ) :
n =
theorem add_smul {R : Type u_2} {M : Type u_5} [] [] [Module R M] (r : R) (s : R) (x : M) :
(r + s) x = r x + s x
theorem Convex.combo_self {R : Type u_2} {M : Type u_5} [] [] [Module R M] {a : R} {b : R} (h : a + b = 1) (x : M) :
a x + b x = x
theorem two_smul (R : Type u_2) {M : Type u_5} [] [] [Module R M] (x : M) :
2 x = x + x
@[deprecated]
theorem two_smul' (R : Type u_2) {M : Type u_5} [] [] [Module R M] (x : M) :
2 x = bit0 x
@[simp]
theorem invOf_two_smul_add_invOf_two_smul (R : Type u_2) {M : Type u_5} [] [] [Module R M] [] (x : M) :
2 x + 2 x = x
@[reducible]
def Function.Injective.module (R : Type u_2) {M : Type u_5} {M₂ : Type u_6} [] [] [Module R M] [] [SMul R M₂] (f : M₂ →+ M) (hf : ) (smul : ∀ (c : R) (x : M₂), f (c x) = c f x) :
Module R M₂

Pullback a Module structure along an injective additive monoid homomorphism. See note [reducible non-instances].

Instances For
def Function.Surjective.module (R : Type u_2) {M : Type u_5} {M₂ : Type u_6} [] [] [Module R M] [] [SMul R M₂] (f : M →+ M₂) (hf : ) (smul : ∀ (c : R) (x : M), f (c x) = c f x) :
Module R M₂

Pushforward a Module structure along a surjective additive monoid homomorphism.

Instances For
@[reducible]
def Function.Surjective.moduleLeft {R : Type u_9} {S : Type u_10} {M : Type u_11} [] [] [Module R M] [] [SMul S M] (f : R →+* S) (hf : ) (hsmul : ∀ (c : R) (x : M), f c x = c x) :
Module S M

Push forward the action of R on M along a compatible surjective map f : R →+* S.

See also Function.Surjective.mulActionLeft and Function.Surjective.distribMulActionLeft.

Instances For
@[reducible]
def Module.compHom {R : Type u_2} {S : Type u_4} (M : Type u_5) [] [] [Module R M] [] (f : S →+* R) :
Module S M

Compose a Module with a RingHom, with action f s • m.

See note [reducible non-instances].

Instances For
@[simp]
theorem Module.toAddMonoidEnd_apply_apply (R : Type u_2) (M : Type u_5) [] [] [Module R M] (x : R) :
∀ (x : M), ↑(↑() x) x = x x
def Module.toAddMonoidEnd (R : Type u_2) (M : Type u_5) [] [] [Module R M] :

(•) as an AddMonoidHom.

This is a stronger version of DistribMulAction.toAddMonoidEnd

Instances For
def smulAddHom (R : Type u_2) (M : Type u_5) [] [] [Module R M] :
R →+ M →+ M

A convenience alias for Module.toAddMonoidEnd as an AddMonoidHom, usually to allow the use of AddMonoidHom.flip.

Instances For
@[simp]
theorem smulAddHom_apply {R : Type u_2} {M : Type u_5} [] [] [Module R M] (r : R) (x : M) :
↑(↑() r) x = r x
theorem Module.eq_zero_of_zero_eq_one {R : Type u_2} {M : Type u_5} [] [] [Module R M] (x : M) (zero_eq_one : 0 = 1) :
x = 0
@[simp]
theorem smul_add_one_sub_smul {M : Type u_5} [] {R : Type u_9} [Ring R] [Module R M] {r : R} {m : M} :
r m + (1 - r) m = m
instance AddCommGroup.intModule (M : Type u_5) [] :
theorem AddMonoid.End.int_cast_def (M : Type u_5) [] (z : ) :
z =
structure Module.Core (R : Type u_2) (M : Type u_5) [] [] extends :
Type (max u_2 u_5)
• smul : RMM
• smul_add : ∀ (r : R) (x y : M), r (x + y) = r x + r y

Scalar multiplication distributes over addition from the left.

• add_smul : ∀ (r s_1 : R) (x : M), (r + s_1) x = r x + s_1 x

Scalar multiplication distributes over addition from the right.

• mul_smul : ∀ (r s_1 : R) (x : M), (r * s_1) x = r s_1 x

Scalar multiplication distributes over multiplication from the right.

• one_smul : ∀ (x : M), 1 x = x

Scalar multiplication by one is the identity.

A structure containing most informations as in a module, except the fields zero_smul and smul_zero. As these fields can be deduced from the other ones when M is an AddCommGroup, this provides a way to construct a module structure by checking less properties, in Module.ofCore.

Instances For
def Module.ofCore {R : Type u_2} {M : Type u_5} [] [] (H : ) :
Module R M

Define Module without proving zero_smul and smul_zero by using an auxiliary structure Module.Core, when the underlying space is an AddCommGroup.

Instances For
theorem Convex.combo_eq_smul_sub_add {R : Type u_2} {M : Type u_5} [] [] [Module R M] {x : M} {y : M} {a : R} {b : R} (h : a + b = 1) :
a x + b y = b (y - x) + x
theorem Module.ext' {R : Type u_9} [] {M : Type u_10} [] (P : Module R M) (Q : Module R M) (w : ∀ (r : R) (m : M), r m = r m) :
P = Q

A variant of Module.ext that's convenient for term-mode.

@[simp]
theorem neg_smul {R : Type u_2} {M : Type u_5} [Ring R] [] [Module R M] (r : R) (x : M) :
-r x = -(r x)
theorem neg_smul_neg {R : Type u_2} {M : Type u_5} [Ring R] [] [Module R M] (r : R) (x : M) :
-r -x = r x
@[simp]
theorem Units.neg_smul {R : Type u_2} {M : Type u_5} [Ring R] [] [Module R M] (u : Rˣ) (x : M) :
-u x = -(u x)
theorem neg_one_smul (R : Type u_2) {M : Type u_5} [Ring R] [] [Module R M] (x : M) :
-1 x = -x
theorem sub_smul {R : Type u_2} {M : Type u_5} [Ring R] [] [Module R M] (r : R) (s : R) (y : M) :
(r - s) y = r y - s y
@[reducible]
def Module.addCommMonoidToAddCommGroup (R : Type u_2) {M : Type u_5} [Ring R] [] [Module R M] :

An AddCommMonoid that is a Module over a Ring carries a natural AddCommGroup structure. See note [reducible non-instances].

Instances For
theorem Module.subsingleton (R : Type u_9) (M : Type u_10) [] [] [] [Module R M] :

A module over a Subsingleton semiring is a Subsingleton. We cannot register this as an instance because Lean has no way to guess R.

theorem Module.nontrivial (R : Type u_9) (M : Type u_10) [] [] [] [Module R M] :

A semiring is Nontrivial provided that there exists a nontrivial module over this semiring.

instance Semiring.toModule {R : Type u_2} [] :
Module R R
instance Semiring.toOppositeModule {R : Type u_2} [] :

Like Semiring.toModule, but multiplies on the right.

def RingHom.toModule {R : Type u_2} {S : Type u_4} [] [] (f : R →+* S) :
Module R S

A ring homomorphism f : R →+* M defines a module structure by r • x = f r * x.

Instances For

The tautological action by R →+* R on R.

This generalizes Function.End.applyMulAction.

@[simp]
theorem RingHom.smul_def {R : Type u_2} [] (f : R →+* R) (a : R) :
f a = f a
instance RingHom.applyFaithfulSMul {R : Type u_2} [] :

RingHom.applyDistribMulAction is faithful.

theorem nsmul_eq_smul_cast (R : Type u_2) {M : Type u_5} [] [] [Module R M] (n : ) (b : M) :
n b = n b

nsmul is equal to any other module structure via a cast.

theorem nat_smul_eq_nsmul {M : Type u_5} [] (h : ) (n : ) (x : M) :
= n x

Convert back any exotic ℕ-smul to the canonical instance. This should not be needed since in mathlib all AddCommMonoids should normally have exactly one ℕ-module structure by design.

All ℕ-module structures are equal. Not an instance since in mathlib all AddCommMonoid should normally have exactly one ℕ-module structure by design.

Instances For
instance AddCommMonoid.nat_isScalarTower {R : Type u_2} {M : Type u_5} [] [] [Module R M] :
theorem zsmul_eq_smul_cast (R : Type u_2) {M : Type u_5} [Ring R] [] [Module R M] (n : ) (b : M) :
n b = n b

zsmul is equal to any other module structure via a cast.

theorem int_smul_eq_zsmul {M : Type u_5} [] (h : ) (n : ) (x : M) :
= n x

Convert back any exotic ℤ-smul to the canonical instance. This should not be needed since in mathlib all AddCommGroups should normally have exactly one ℤ-module structure by design.

All ℤ-module structures are equal. Not an instance since in mathlib all AddCommGroup should normally have exactly one ℤ-module structure by design.

Instances For
theorem map_int_cast_smul {M : Type u_5} {M₂ : Type u_6} [] [] {F : Type u_9} [] (f : F) (R : Type u_10) (S : Type u_11) [Ring R] [Ring S] [Module R M] [Module S M₂] (x : ) (a : M) :
f (x a) = x f a
theorem map_nat_cast_smul {M : Type u_5} {M₂ : Type u_6} [] [] {F : Type u_9} [] (f : F) (R : Type u_10) (S : Type u_11) [] [] [Module R M] [Module S M₂] (x : ) (a : M) :
f (x a) = x f a
theorem map_inv_nat_cast_smul {M : Type u_5} {M₂ : Type u_6} [] [] {F : Type u_9} [] (f : F) (R : Type u_10) (S : Type u_11) [] [] [Module R M] [Module S M₂] (n : ) (x : M) :
f ((n)⁻¹ x) = (n)⁻¹ f x
theorem map_inv_int_cast_smul {M : Type u_5} {M₂ : Type u_6} [] [] {F : Type u_9} [] (f : F) (R : Type u_10) (S : Type u_11) [] [] [Module R M] [Module S M₂] (z : ) (x : M) :
f ((z)⁻¹ x) = (z)⁻¹ f x
theorem map_rat_cast_smul {M : Type u_5} {M₂ : Type u_6} [] [] {F : Type u_9} [] (f : F) (R : Type u_10) (S : Type u_11) [] [] [Module R M] [Module S M₂] (c : ) (x : M) :
f (c x) = c f x
theorem map_rat_smul {M : Type u_5} {M₂ : Type u_6} [] [] [] [Module M₂] {F : Type u_9} [] (f : F) (c : ) (x : M) :
f (c x) = c f x
instance subsingleton_rat_module (E : Type u_9) [] :

There can be at most one Module ℚ E structure on an additive commutative group.

theorem inv_nat_cast_smul_eq {E : Type u_9} (R : Type u_10) (S : Type u_11) [] [] [] [Module R E] [Module S E] (n : ) (x : E) :
(n)⁻¹ x = (n)⁻¹ x

If E is a vector space over two division semirings R and S, then scalar multiplications agree on inverses of natural numbers in R and S.

theorem inv_int_cast_smul_eq {E : Type u_9} (R : Type u_10) (S : Type u_11) [] [] [] [Module R E] [Module S E] (n : ) (x : E) :
(n)⁻¹ x = (n)⁻¹ x

If E is a vector space over two division rings R and S, then scalar multiplications agree on inverses of integer numbers in R and S.

theorem inv_nat_cast_smul_comm {α : Type u_9} {E : Type u_10} (R : Type u_11) [] [] [] [Module R E] [] (n : ) (s : α) (x : E) :
(n)⁻¹ s x = s (n)⁻¹ x

If E is a vector space over a division semiring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of natural numbers in R.

theorem inv_int_cast_smul_comm {α : Type u_9} {E : Type u_10} (R : Type u_11) [] [] [] [Module R E] [] (n : ) (s : α) (x : E) :
(n)⁻¹ s x = s (n)⁻¹ x

If E is a vector space over a division ring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of integers in R

theorem rat_cast_smul_eq {E : Type u_9} (R : Type u_10) (S : Type u_11) [] [] [] [Module R E] [Module S E] (r : ) (x : E) :
r x = r x

If E is a vector space over two division rings R and S, then scalar multiplications agree on rational numbers in R and S.

instance AddCommGroup.intIsScalarTower {R : Type u} {M : Type v} [Ring R] [] [Module R M] :
instance IsScalarTower.rat {R : Type u} {M : Type v} [Ring R] [] [Module R M] [] [] :
instance SMulCommClass.rat {R : Type u} {M : Type v} [] [] [Module R M] [] :
instance SMulCommClass.rat' {R : Type u} {M : Type v} [] [] [Module R M] [] :

### NoZeroSMulDivisors#

This section defines the NoZeroSMulDivisors class, and includes some tests for the vanishing of elements (especially in modules over division rings).

class NoZeroSMulDivisors (R : Type u_9) (M : Type u_10) [Zero R] [Zero M] [SMul R M] :
• eq_zero_or_eq_zero_of_smul_eq_zero : ∀ {c : R} {x : M}, c x = 0c = 0 x = 0

If scalar multiplication yields zero, either the scalar or the vector was zero.

NoZeroSMulDivisors R M states that a scalar multiple is 0 only if either argument is 0. This is a version of saying that M is torsion free, without assuming R is zero-divisor free.

The main application of NoZeroSMulDivisors R M, when M is a module, is the result smul_eq_zero: a scalar multiple is 0 iff either argument is 0.

It is a generalization of the NoZeroDivisors class to heterogeneous multiplication.

Instances
theorem Function.Injective.noZeroSMulDivisors {R : Type u_9} {M : Type u_10} {N : Type u_11} [Zero R] [Zero M] [Zero N] [SMul R M] [SMul R N] [] (f : MN) (hf : ) (h0 : f 0 = 0) (hs : ∀ (c : R) (x : M), f (c x) = c f x) :

Pullback a NoZeroSMulDivisors instance along an injective function.

instance NoZeroDivisors.toNoZeroSMulDivisors {R : Type u_2} [Zero R] [Mul R] [] :
theorem smul_ne_zero {R : Type u_2} {M : Type u_5} [Zero R] [Zero M] [SMul R M] [] {c : R} {x : M} (hc : c 0) (hx : x 0) :
c x 0
@[simp]
theorem smul_eq_zero {R : Type u_2} {M : Type u_5} [Zero R] [Zero M] [] [] {c : R} {x : M} :
c x = 0 c = 0 x = 0
theorem smul_ne_zero_iff {R : Type u_2} {M : Type u_5} [Zero R] [Zero M] [] [] {c : R} {x : M} :
c x 0 c 0 x 0
theorem Nat.noZeroSMulDivisors (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] :
theorem two_nsmul_eq_zero (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] {v : M} :
2 v = 0 v = 0
theorem CharZero.of_module (R : Type u_2) [] (M : Type u_9) [] [Module R M] :

If M is an R-module with one and M has characteristic zero, then R has characteristic zero as well. Usually M is an R-algebra.

theorem smul_right_injective {R : Type u_2} (M : Type u_5) [] [] [Module R M] [] {c : R} (hc : c 0) :
Function.Injective ((fun x x_1 => x x_1) c)
theorem smul_right_inj {R : Type u_2} {M : Type u_5} [] [] [Module R M] [] {c : R} (hc : c 0) {x : M} {y : M} :
c x = c y x = y
theorem self_eq_neg (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] {v : M} :
v = -v v = 0
theorem neg_eq_self (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] {v : M} :
-v = v v = 0
theorem self_ne_neg (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] {v : M} :
v -v v 0
theorem neg_ne_self (R : Type u_2) (M : Type u_5) [] [] [Module R M] [] [] {v : M} :
-v v v 0
theorem smul_left_injective (R : Type u_2) {M : Type u_5} [Ring R] [] [Module R M] [] {x : M} (hx : x 0) :
Function.Injective fun c => c x
instance GroupWithZero.toNoZeroSMulDivisors {R : Type u_2} {M : Type u_5} [] [] [] :

This instance applies to DivisionSemirings, in particular NNReal and NNRat.

instance RatModule.noZeroSMulDivisors {M : Type u_5} [] [] :
theorem Nat.smul_one_eq_coe {R : Type u_9} [] (m : ) :
m 1 = m
theorem Int.smul_one_eq_coe {R : Type u_9} [Ring R] (m : ) :
m 1 = m