Abstract base class for modules#

AUTHORS:

William Stein: initial version
Julian Rueth (2014-05-10): category parameter for Module, doc cleanup

EXAMPLES:

A minimal example of a module:

Sage

sage: from sage.structure.richcmp import richcmp
sage: class MyElement(sage.structure.element.ModuleElement):
....:     def __init__(self, parent, x):
....:         self.x = x
....:         sage.structure.element.ModuleElement.__init__(self, parent=parent)
....:     def _lmul_(self, c):
....:         return self.parent()(c*self.x)
....:     def _add_(self, other):
....:         return self.parent()(self.x + other.x)
....:     def _richcmp_(self, other, op):
....:         return richcmp(self.x, other.x, op)
....:     def __hash__(self):
....:         return hash(self.x)
....:     def _repr_(self):
....:         return repr(self.x)

sage: from sage.modules.module import Module
sage: class MyModule(Module):
....:     Element = MyElement
....:     def _element_constructor_(self, x):
....:         if isinstance(x, MyElement): x = x.x
....:         return self.element_class(self, self.base_ring()(x))
....:     def __eq__(self, other):
....:         if not isinstance(other, MyModule): return False
....:         return self.base_ring() == other.base_ring()
....:     def __hash__(self):
....:         return hash(self.base_ring())

sage: M = MyModule(QQ)
sage: M(1)
1

sage: import __main__
sage: __main__.MyModule = MyModule
sage: __main__.MyElement = MyElement
sage: TestSuite(M).run()

Python

>>> from sage.all import *
>>> from sage.structure.richcmp import richcmp
>>> class MyElement(sage.structure.element.ModuleElement):
...     def __init__(self, parent, x):
...         self.x = x
...         sage.structure.element.ModuleElement.__init__(self, parent=parent)
...     def _lmul_(self, c):
...         return self.parent()(c*self.x)
...     def _add_(self, other):
...         return self.parent()(self.x + other.x)
...     def _richcmp_(self, other, op):
...         return richcmp(self.x, other.x, op)
...     def __hash__(self):
...         return hash(self.x)
...     def _repr_(self):
...         return repr(self.x)

>>> from sage.modules.module import Module
>>> class MyModule(Module):
...     Element = MyElement
...     def _element_constructor_(self, x):
...         if isinstance(x, MyElement): x = x.x
...         return self.element_class(self, self.base_ring()(x))
...     def __eq__(self, other):
...         if not isinstance(other, MyModule): return False
...         return self.base_ring() == other.base_ring()
...     def __hash__(self):
...         return hash(self.base_ring())

>>> M = MyModule(QQ)
>>> M(Integer(1))
1

>>> import __main__
>>> __main__.MyModule = MyModule
>>> __main__.MyElement = MyElement
>>> TestSuite(M).run()

class sage.modules.module.Module[source]#

Bases: Parent

Generic module class.

INPUT:

base – a ring. The base ring of the module.
category – a category (default: None), the category for this module. If None, then this is set to the category of modules/vector spaces over base.
names – names of generators

EXAMPLES:

Sage

sage: from sage.modules.module import Module
sage: M = Module(ZZ)
sage: M.base_ring()
Integer Ring
sage: M.category()
Category of modules over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modules.module import Module
>>> M = Module(ZZ)
>>> M.base_ring()
Integer Ring
>>> M.category()
Category of modules over Integer Ring

Normally the category is set to the category of modules over base. If base is a field, then the category is the category of vector spaces over base:

Sage

sage: M_QQ = Module(QQ)
sage: M_QQ.category()
Category of vector spaces over Rational Field

Python

>>> from sage.all import *
>>> M_QQ = Module(QQ)
>>> M_QQ.category()
Category of vector spaces over Rational Field

The category parameter can be used to set a more specific category:

Sage

sage: N = Module(ZZ, category=FiniteDimensionalModulesWithBasis(ZZ))
sage: N.category()
Category of finite dimensional modules with basis over Integer Ring

Python

>>> from sage.all import *
>>> N = Module(ZZ, category=FiniteDimensionalModulesWithBasis(ZZ))
>>> N.category()
Category of finite dimensional modules with basis over Integer Ring

base_extend(R)[source]#

Return the base extension of self to \(R\).

This is the same as self.change_ring(R) except that a TypeError is raised if there is no canonical coerce map from the base ring of self to \(R\).

INPUT:

R – ring

EXAMPLES:

Sage

sage: V = ZZ^7                                                              # needs sage.modules
sage: V.base_extend(QQ)                                                     # needs sage.modules
Vector space of dimension 7 over Rational Field

Python

>>> from sage.all import *
>>> V = ZZ**Integer(7)                                                              # needs sage.modules
>>> V.base_extend(QQ)                                                     # needs sage.modules
Vector space of dimension 7 over Rational Field

change_ring(R)[source]#

Return the base change of self to \(R\).

EXAMPLES:

Sage

sage: from sage.modular.modform.space import ModularFormsSpace              # needs sage.modular
sage: ModularFormsSpace(Gamma0(11), 2,                                      # needs sage.modular sage.rings.finite_rings
....:                   DirichletGroup(1)[0], QQ).change_ring(GF(7))
Traceback (most recent call last):
...
NotImplementedError: the method change_ring() has not yet been implemented

Python

>>> from sage.all import *
>>> from sage.modular.modform.space import ModularFormsSpace              # needs sage.modular
>>> ModularFormsSpace(Gamma0(Integer(11)), Integer(2),                                      # needs sage.modular sage.rings.finite_rings
...                   DirichletGroup(Integer(1))[Integer(0)], QQ).change_ring(GF(Integer(7)))
Traceback (most recent call last):
...
NotImplementedError: the method change_ring() has not yet been implemented

endomorphism_ring()[source]#

Return the endomorphism ring of this module in its category.

EXAMPLES:

Sage

sage: from sage.modules.module import Module
sage: M = Module(ZZ)
sage: M.endomorphism_ring()
Set of Morphisms
 from <sage.modules.module.Module object at ...>
   to <sage.modules.module.Module object at ...>
   in Category of modules over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modules.module import Module
>>> M = Module(ZZ)
>>> M.endomorphism_ring()
Set of Morphisms
 from <sage.modules.module.Module object at ...>
   to <sage.modules.module.Module object at ...>
   in Category of modules over Integer Ring

sage.modules.module.is_Module(x)[source]#

Return True if x is a module, False otherwise.

INPUT:

x – anything.

EXAMPLES:

Sage

sage: from sage.modules.module import is_Module
sage: M = FreeModule(RationalField(),30)                                        # needs sage.modules
sage: is_Module(M)                                                              # needs sage.modules
doctest:warning...
DeprecationWarning: the function is_Module is deprecated;
use 'isinstance(..., Module)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
sage: is_Module(10)
False

Python

>>> from sage.all import *
>>> from sage.modules.module import is_Module
>>> M = FreeModule(RationalField(),Integer(30))                                        # needs sage.modules
>>> is_Module(M)                                                              # needs sage.modules
doctest:warning...
DeprecationWarning: the function is_Module is deprecated;
use 'isinstance(..., Module)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
>>> is_Module(Integer(10))
False

sage.modules.module.is_VectorSpace(x)[source]#

Return True if x is a vector space, False otherwise.

INPUT:

x – anything.

EXAMPLES:

Sage

sage: # needs sage.modules
sage: from sage.modules.module import is_Module, is_VectorSpace
sage: M = FreeModule(RationalField(),30)
sage: is_VectorSpace(M)
doctest:warning...
DeprecationWarning: the function is_VectorSpace is deprecated;
use 'isinstance(..., Module)' and check the base ring instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
sage: M = FreeModule(IntegerRing(),30)
sage: is_Module(M)
doctest:warning...
DeprecationWarning: the function is_Module is deprecated;
use 'isinstance(..., Module)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
sage: is_VectorSpace(M)
False

Python

>>> from sage.all import *
>>> # needs sage.modules
>>> from sage.modules.module import is_Module, is_VectorSpace
>>> M = FreeModule(RationalField(),Integer(30))
>>> is_VectorSpace(M)
doctest:warning...
DeprecationWarning: the function is_VectorSpace is deprecated;
use 'isinstance(..., Module)' and check the base ring instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
>>> M = FreeModule(IntegerRing(),Integer(30))
>>> is_Module(M)
doctest:warning...
DeprecationWarning: the function is_Module is deprecated;
use 'isinstance(..., Module)' instead
See https://github.com/sagemath/sage/issues/37924 for details.
True
>>> is_VectorSpace(M)
False