Algebras

class sage.categories.algebras.Algebras(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring, sage.categories.cartesian_product.CategoryWithCartesianProduct

The category of algebras over a given base ring.

An algebra over a ring R is a module over R which is itself a ring.

TODO: should R be a commutative ring?

EXAMPLES:

sage: Algebras(ZZ)
Category of algebras over Integer Ring
sage: Algebras(ZZ).super_categories()
[Category of rings, Category of modules over Integer Ring]

TESTS:

sage: TestSuite(Algebras(ZZ)).run()
class CartesianProductCategory(category, name=None)

Bases: sage.categories.cartesian_product.CartesianProductCategory

The category of algebras constructed by cartesian products of algebras

See:
ParentMethods
alias of CartesianProductCategory.ParentMethods
super_categories(*args, **kwds)

EXAMPLES:

sage: Algebras(QQ).cartesian_product_category().super_categories()
[Category of algebras over Rational Field]
class Algebras.ElementMethods
class Algebras.ParentMethods
from_base_ring(r)

Canonical embedding from base ring

INPUT:

  • r – an element of self.base_ring()

Returns the canonical embedding of r into self.

EXAMPLES:

sage: A = AlgebrasWithBasis(QQ).example(); A
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: A.from_base_ring(1)
B[word: ]
Algebras.dual()

Returns the dual category

EXAMPLES:

The category of algebras over the Rational Field is dual to the category of coalgebras over the same field:

sage: C = Algebras(QQ)
sage: C.dual()
Category of coalgebras over Rational Field
Algebras.super_categories(*args, **kwds)

EXAMPLES:

sage: Algebras(QQ).super_categories()
[Category of rings, Category of modules over Rational Field]

Previous topic

AlgebraModules

Next topic

AlgebrasWithBasis

This Page