# Modules¶

class sage.categories.modules.Modules(base, name=None)

The category of all modules over a base ring $$R$$.

An $$R$$-module $$M$$ is a left and right $$R$$-module over a commutative ring $$R$$ such that:

$r*(x*s) = (r*x)*s \qquad \forall r,s \in R \text{ and } x \in M$

INPUT:

• base_ring – a ring $$R$$ or subcategory of Rings()
• dispatch – a boolean (for internal use; default: True)

When the base ring is a field, the category of vector spaces is returned instead (unless dispatch == False).

Warning

Outside of the context of symmetric modules over a commutative ring, the specifications of this category are fuzzy and not yet set in stone (see below). The code in this category and its subcategories is therefore prone to bugs or arbitrary limitations in this case.

EXAMPLES:

sage: Modules(ZZ)
Category of modules over Integer Ring
sage: Modules(QQ)
Category of vector spaces over Rational Field

sage: Modules(Rings())
Category of modules over rings
sage: Modules(FiniteFields())
Category of vector spaces over finite enumerated fields

sage: Modules(Integers(9))
Category of modules over Ring of integers modulo 9

sage: Modules(Integers(9)).super_categories()
[Category of bimodules over Ring of integers modulo 9 on the left and Ring of integers modulo 9 on the right]

sage: Modules(ZZ).super_categories()
[Category of bimodules over Integer Ring on the left and Integer Ring on the right]

sage: Modules == RingModules
True

sage: Modules(ZZ['x']).is_abelian()   # see #6081
True


Todo

• Clarify the distinction, if any, with BiModules(R, R). In particular, if $$R$$ is a commutative ring (e.g. a field), some pieces of the code possibly assume that $$M$$ is a symmetric R-R-bimodule:

$r*x = x*r \qquad \forall r \in R \text{ and } x \in M$
• Make sure that non symmetric modules are properly supported by all the code, and advertise it.

• Make sure that non commutative rings are properly supported by all the code, and advertise it.

• Add support for base semirings.

• Implement a FreeModules(R) category, when so prompted by a concrete use case: e.g. modeling a free module with several bases (using Sets.SubcategoryMethods.Realizations()) or with an atlas of local maps (see e.g. trac ticket #15916).

class CartesianProducts(category, *args)

The category of modules constructed as Cartesian products of modules

This construction gives the direct product of modules. The implementation is based on the following resources:

class ParentMethods
base_ring()

Return the base ring of this Cartesian product.

EXAMPLES:

sage: E = CombinatorialFreeModule(ZZ, [1,2,3])
sage: F = CombinatorialFreeModule(ZZ, [2,3,4])
sage: C = cartesian_product([E, F]); C
Free module generated by {1, 2, 3} over Integer Ring (+)
Free module generated by {2, 3, 4} over Integer Ring
sage: C.base_ring()
Integer Ring

extra_super_categories()

A Cartesian product of modules is endowed with a natural module structure.

EXAMPLES:

sage: Modules(ZZ).CartesianProducts().extra_super_categories()
[Category of modules over Integer Ring]
sage: Modules(ZZ).CartesianProducts().super_categories()
[Category of Cartesian products of commutative additive groups,
Category of modules over Integer Ring]

class ElementMethods
Filtered
class FiniteDimensional(base_category)
extra_super_categories()

Implement the fact that a finite dimensional module over a finite ring is finite.

EXAMPLES:

sage: Modules(IntegerModRing(4)).FiniteDimensional().extra_super_categories()
[Category of finite sets]
sage: Modules(ZZ).FiniteDimensional().extra_super_categories()
[]
sage: Modules(GF(5)).FiniteDimensional().is_subcategory(Sets().Finite())
True
sage: Modules(ZZ).FiniteDimensional().is_subcategory(Sets().Finite())
False

sage: Modules(Rings().Finite()).FiniteDimensional().is_subcategory(Sets().Finite())
True
sage: Modules(Rings()).FiniteDimensional().is_subcategory(Sets().Finite())
False

Graded
class Homsets(category, *args)

The category of homomorphism sets $$\hom(X,Y)$$ for $$X$$, $$Y$$ modules.

class Endset(base_category)

The category of endomorphism sets $$End(X)$$ for $$X$$ a module (this is not used yet)

extra_super_categories()

Implement the fact that the endomorphism set of a module is an algebra.

CategoryWithAxiom.extra_super_categories()

EXAMPLES:

sage: Modules(ZZ).Endsets().extra_super_categories()
[Category of magmatic algebras over Integer Ring]

sage: End(ZZ^3) in Algebras(ZZ)
True

class ParentMethods
base_ring()

Return the base ring of self.

EXAMPLES:

sage: E = CombinatorialFreeModule(ZZ, [1,2,3])
sage: F = CombinatorialFreeModule(ZZ, [2,3,4])
sage: H = Hom(E, F)
sage: H.base_ring()
Integer Ring


This base_ring method is actually overridden by sage.structure.category_object.CategoryObject.base_ring():

sage: H.base_ring.__module__


Here we call it directly:

sage: method = H.category().parent_class.base_ring
sage: method.__get__(H)()
Integer Ring

zero()

EXAMPLES:

sage: E = CombinatorialFreeModule(ZZ, [1,2,3])
sage: F = CombinatorialFreeModule(ZZ, [2,3,4])
sage: H = Hom(E, F)
sage: f = H.zero()
sage: f
Generic morphism:
From: Free module generated by {1, 2, 3} over Integer Ring
To:   Free module generated by {2, 3, 4} over Integer Ring
sage: f(E.monomial(2))
0
sage: f(E.monomial(3)) == F.zero()
True

base_ring()

EXAMPLES:

sage: Modules(ZZ).Homsets().base_ring()
Integer Ring


Todo

Generalize this so that any homset category of a full subcategory of modules over a base ring is a category over this base ring.

extra_super_categories()

EXAMPLES:

sage: Modules(ZZ).Homsets().extra_super_categories()
[Category of modules over Integer Ring]

class ParentMethods
tensor_square()

Returns the tensor square of self

EXAMPLES:

sage: A = HopfAlgebrasWithBasis(QQ).example()
sage: A.tensor_square()
An example of Hopf algebra with basis:
the group algebra of the Dihedral group of order 6
as a permutation group over Rational Field # An example
of Hopf algebra with basis: the group algebra of the Dihedral
group of order 6 as a permutation group over Rational Field

class SubcategoryMethods
DualObjects()

Return the category of spaces constructed as duals of spaces of self.

The dual of a vector space $$V$$ is the space consisting of all linear functionals on $$V$$ (see Wikipedia article Dual_space). Additional structure on $$V$$ can endow its dual with additional structure; for example, if $$V$$ is a finite dimensional algebra, then its dual is a coalgebra.

This returns the category of spaces constructed as dual of spaces in self, endowed with the appropriate additional structure.

Warning

• This semantic of dual and DualObject is imposed on all subcategories, in particular to make dual a covariant functorial construction.

A subcategory that defines a different notion of dual needs to use a different name.

• Typically, the category of graded modules should define a separate graded_dual construction (see trac ticket #15647). For now the two constructions are not distinguished which is an oversimplified model.

EXAMPLES:

sage: VectorSpaces(QQ).DualObjects()
Category of duals of vector spaces over Rational Field


The dual of a vector space is a vector space:

sage: VectorSpaces(QQ).DualObjects().super_categories()
[Category of vector spaces over Rational Field]


The dual of an algebra is a coalgebra:

sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str)
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]


The dual of a coalgebra is an algebra:

sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str)
[Category of algebras over Rational Field,
Category of duals of vector spaces over Rational Field]


As a shorthand, this category can be accessed with the dual() method:

sage: VectorSpaces(QQ).dual()
Category of duals of vector spaces over Rational Field

Filtered(base_ring=None)

Return the subcategory of the filtered objects of self.

INPUT:

• base_ring – this is ignored

EXAMPLES:

sage: Modules(ZZ).Filtered()
Category of filtered modules over Integer Ring

sage: Coalgebras(QQ).Filtered()
Join of Category of filtered modules over Rational Field
and Category of coalgebras over Rational Field

sage: AlgebrasWithBasis(QQ).Filtered()
Category of filtered algebras with basis over Rational Field


Todo

• Explain why this does not commute with WithBasis()
• Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the base_ring argument.
FiniteDimensional()

Return the full subcategory of the finite dimensional objects of self.

EXAMPLES:

sage: Modules(ZZ).FiniteDimensional()
Category of finite dimensional modules over Integer Ring
sage: Coalgebras(QQ).FiniteDimensional()
Category of finite dimensional coalgebras over Rational Field
sage: AlgebrasWithBasis(QQ).FiniteDimensional()
Category of finite dimensional algebras with basis over Rational Field

Graded(base_ring=None)

Return the subcategory of the graded objects of self.

INPUT:

• base_ring – this is ignored

EXAMPLES:

sage: Modules(ZZ).Graded()
Category of graded modules over Integer Ring

Join of Category of graded modules over Rational Field and Category of coalgebras over Rational Field

Category of graded algebras with basis over Rational Field


Todo

• Explain why this does not commute with WithBasis()
• Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the base_ring argument.
Super(base_ring=None)

Return the super-analogue category of self.

INPUT:

• base_ring – this is ignored

EXAMPLES:

sage: Modules(ZZ).Super()
Category of super modules over Integer Ring

sage: Coalgebras(QQ).Super()
Category of super coalgebras over Rational Field

sage: AlgebrasWithBasis(QQ).Super()
Category of super algebras with basis over Rational Field


Todo

• Explain why this does not commute with WithBasis()
• Improve the support for covariant functorial constructions categories over a base ring so as to get rid of the base_ring argument.
TensorProducts()

Return the full subcategory of objects of self constructed as tensor products.

EXAMPLES:

sage: ModulesWithBasis(QQ).TensorProducts()
Category of tensor products of vector spaces with basis over Rational Field

WithBasis()

Return the full subcategory of the objects of self with a distinguished basis.

EXAMPLES:

sage: Modules(ZZ).WithBasis()
Category of modules with basis over Integer Ring
sage: Coalgebras(QQ).WithBasis()
Category of coalgebras with basis over Rational Field
sage: AlgebrasWithBasis(QQ).WithBasis()
Category of algebras with basis over Rational Field

base_ring()

Return the base ring (category) for self.

This implements a base_ring method for all subcategories of Modules(K).

EXAMPLES:

sage: C = Modules(QQ) & Semigroups(); C
Join of Category of semigroups and Category of vector spaces over Rational Field
sage: C.base_ring()
Rational Field
sage: C.base_ring.__module__
'sage.categories.modules'

sage: C = Modules(Rings()) & Semigroups(); C
Join of Category of semigroups and Category of modules over rings
sage: C.base_ring()
Category of rings
sage: C.base_ring.__module__
'sage.categories.modules'

sage: C = DescentAlgebra(QQ,3).B().category()
sage: C.base_ring.__module__
'sage.categories.modules'
sage: C.base_ring()
Rational Field

sage: C = QuasiSymmetricFunctions(QQ).F().category()
sage: C.base_ring.__module__
'sage.categories.modules'
sage: C.base_ring()
Rational Field

dual()

Return the category of spaces constructed as duals of spaces of self.

The dual of a vector space $$V$$ is the space consisting of all linear functionals on $$V$$ (see Wikipedia article Dual_space). Additional structure on $$V$$ can endow its dual with additional structure; for example, if $$V$$ is a finite dimensional algebra, then its dual is a coalgebra.

This returns the category of spaces constructed as dual of spaces in self, endowed with the appropriate additional structure.

Warning

• This semantic of dual and DualObject is imposed on all subcategories, in particular to make dual a covariant functorial construction.

A subcategory that defines a different notion of dual needs to use a different name.

• Typically, the category of graded modules should define a separate graded_dual construction (see trac ticket #15647). For now the two constructions are not distinguished which is an oversimplified model.

EXAMPLES:

sage: VectorSpaces(QQ).DualObjects()
Category of duals of vector spaces over Rational Field


The dual of a vector space is a vector space:

sage: VectorSpaces(QQ).DualObjects().super_categories()
[Category of vector spaces over Rational Field]


The dual of an algebra is a coalgebra:

sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str)
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]


The dual of a coalgebra is an algebra:

sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str)
[Category of algebras over Rational Field,
Category of duals of vector spaces over Rational Field]


As a shorthand, this category can be accessed with the dual() method:

sage: VectorSpaces(QQ).dual()
Category of duals of vector spaces over Rational Field

Super
class TensorProducts(category, *args)

The category of modules constructed by tensor product of modules.

extra_super_categories()

EXAMPLES:

sage: Modules(ZZ).TensorProducts().extra_super_categories()
[Category of modules over Integer Ring]
sage: Modules(ZZ).TensorProducts().super_categories()
[Category of modules over Integer Ring]

WithBasis
additional_structure()

Return None.

Indeed, the category of modules defines no additional structure: a bimodule morphism between two modules is a module morphism.

Todo

Should this category be a CategoryWithAxiom?

EXAMPLES:

sage: Modules(ZZ).additional_structure()

super_categories()

EXAMPLES:

sage: Modules(ZZ).super_categories()
[Category of bimodules over Integer Ring on the left and Integer Ring on the right]


Nota bene:

sage: Modules(QQ)
Category of vector spaces over Rational Field
sage: Modules(QQ).super_categories()
[Category of modules over Rational Field]