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 is a module over R which is itself a ring.
TODO: should 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()
Bases: sage.categories.cartesian_product.CartesianProductCategory
The category of algebras constructed by cartesian products of algebras
EXAMPLES:
sage: Algebras(QQ).cartesian_product_category().super_categories()
[Category of algebras over Rational Field]
Canonical embedding from base ring
INPUT:
- r – an element of self.base_ring()
Returns the canonical embedding of 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: ]
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
EXAMPLES:
sage: Algebras(QQ).super_categories()
[Category of rings, Category of modules over Rational Field]