Commutative algebras - Category Framework

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# Commutative algebras

class sage.categories.commutative_algebras.CommutativeAlgebras(base_category)

  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)

    Bases: `TensorProductsCategory`

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

    extra_super_categories()

      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]