Commutative algebras#

class sage.categories.commutative_algebras.CommutativeAlgebras(base_category)[source]#

Bases: CategoryWithAxiom_over_base_ring

The category of commutative algebras with unit over a given base ring.

EXAMPLES:

Sage

sage: M = CommutativeAlgebras(GF(19))
sage: M
Category of commutative algebras over Finite Field of size 19
sage: CommutativeAlgebras(QQ).super_categories()
[Category of algebras over Rational Field, Category of commutative rings]

Python

>>> from sage.all import *
>>> M = CommutativeAlgebras(GF(Integer(19)))
>>> M
Category of commutative algebras over Finite Field of size 19
>>> CommutativeAlgebras(QQ).super_categories()
[Category of algebras over Rational Field, Category of commutative rings]

This is just a shortcut for:

Sage

sage: Algebras(QQ).Commutative()
Category of commutative algebras over Rational Field

Python

>>> from sage.all import *
>>> Algebras(QQ).Commutative()
Category of commutative algebras over Rational Field

Todo

product ( = Cartesian product)
coproduct ( = tensor product over base ring)

class TensorProducts(category, *args)[source]#

Bases: TensorProductsCategory

The category of commutative algebras constructed by tensor product of commutative algebras.

extra_super_categories()[source]#

EXAMPLES:

Sage

sage: Algebras(QQ).Commutative().TensorProducts().extra_super_categories()
[Category of commutative rings]
sage: Algebras(QQ).Commutative().TensorProducts().super_categories()
[Category of tensor products of algebras over Rational Field,
 Category of commutative algebras over Rational Field]

Python

>>> from sage.all import *
>>> Algebras(QQ).Commutative().TensorProducts().extra_super_categories()
[Category of commutative rings]
>>> Algebras(QQ).Commutative().TensorProducts().super_categories()
[Category of tensor products of algebras over Rational Field,
 Category of commutative algebras over Rational Field]