AUTHORS:
- David Kohel & William Stein (2005): initial revision
- Nicolas M. Thiery (2008): rewrote for new category framework
Bases: sage.categories.category_types.Category_over_base_ring
The category of algebras over a given base ring.
An algebra over a ring is a module over 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.CartesianProductsCategory
The category of algebras constructed as cartesian products of algebras
This construction gives the direct product of algebras. See discussion on:
A cartesian product of algebras is endowed with a natural algebra structure.
EXAMPLES:
sage: Algebras(QQ).CartesianProducts().extra_super_categories()
[Category of algebras over Rational Field]
sage: Algebras(QQ).CartesianProducts().super_categories()
[Category of algebras over Rational Field, Category of Cartesian products of monoids]
Bases: sage.categories.dual.DualObjectsCategory
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 duals of algebras over Rational Field
sage: C.dual().extra_super_categories()
[Category of coalgebras 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: ]
Bases: sage.categories.tensor.TensorProductsCategory
EXAMPLES:
sage: Algebras(QQ).TensorProducts().extra_super_categories()
[Category of algebras over Rational Field]
sage: Algebras(QQ).TensorProducts().super_categories()
[Category of algebras over Rational Field]
Meaning: a tensor product of algebras is an algebra
EXAMPLES:
sage: Algebras(ZZ).super_categories()
[Category of rings, Category of modules over Integer Ring]