Modules over ℕ
and ℤ
#
This file concerns modules where the scalars are the natural numbers or the integers.
Main definitions #
AddCommMonoid.toNatModule
: anyAddCommMonoid
is (uniquely) a module over the naturals.AddCommGroup.toIntModule
: anyAddCommGroup
is a module over the integers.
Main results #
AddCommMonoid.uniqueNatModule
: there is an uniqueAddCommMonoid ℕ M
structure for anyM
Tags #
semimodule, module, vector space
Equations
- AddCommMonoid.toNatModule = { toSMul := AddMonoid.toNatSMul, one_smul := ⋯, mul_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }
Equations
- AddCommGroup.toIntModule M = { toSMul := SubNegMonoid.toZSMul, one_smul := ⋯, mul_smul := ⋯, smul_zero := ⋯, smul_add := ⋯, add_smul := ⋯, zero_smul := ⋯ }
An AddCommMonoid
that is a Module
over a Ring
carries a natural AddCommGroup
structure.
See note [reducible non-instances].
Equations
- One or more equations did not get rendered due to their size.
Instances For
nsmul
is equal to any other module structure via a cast.
Convert back any exotic ℕ
-smul to the canonical instance. This should not be needed since in
mathlib all AddCommMonoid
s 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.
Equations
- AddCommMonoid.uniqueNatModule = { default := inferInstance, uniq := ⋯ }
Instances For
All ℕ
-module structures are equal. See also AddCommMoniod.uniqueNatModule
.
Convert back any exotic ℤ
-smul to the canonical instance. This should not be needed since in
mathlib all AddCommGroup
s 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.
Equations
- AddCommGroup.uniqueIntModule = { default := inferInstance, uniq := ⋯ }
Instances For
All ℤ
-module structures are equal. See also AddCommGroup.uniqueIntModule
.