Division rings - Category Framework

Skip to content
Back to top
View this page
Edit this page

# Division rings

class sage.categories.division_rings.DivisionRings(base_category)

  Bases: `CategoryWithAxiom_singleton`

  The category of division rings.

  A division ring (or skew field) is a not necessarily commutative ring where all nonzero elements have multiplicative inverses

  EXAMPLES:

  Sage

      sage: DivisionRings()
      Category of division rings
      sage: DivisionRings().super_categories()
      [Category of domains]

  Python

      >>> from sage.all import *
      >>> DivisionRings()
      Category of division rings
      >>> DivisionRings().super_categories()
      [Category of domains]

  Commutative

    alias of `Fields`

  class ElementMethods

    Bases: `object`

  Finite_extra_super_categories()

    Return extraneous super categories for `DivisionRings().Finite()`.

    EXAMPLES:

    Any field is a division ring:

    Sage

        sage: Fields().is_subcategory(DivisionRings())
        True

    Python

        >>> from sage.all import *
        >>> Fields().is_subcategory(DivisionRings())
        True

    This methods specifies that, by Weddeburn theorem, the reciprocal holds in the finite case: a finite division ring is commutative and thus a field:

    Sage

        sage: DivisionRings().Finite_extra_super_categories()
        (Category of commutative magmas,)
        sage: DivisionRings().Finite()
        Category of finite enumerated fields

    [Python]

        >>> from sage.all import *
        >>> DivisionRings().Finite_extra_super_categories()
        (Category of commutative magmas,)
        >>> DivisionRings().Finite()
        Category of finite enumerated fields

    Warning

    This is not implemented in `DivisionRings.Finite.extra_super_categories` because the categories of finite division rings and of finite fields coincide. See the section Deduction rules in the documentation of axioms.

  class ParentMethods

    Bases: `object`

  extra_super_categories()

    Return the `Domains` category.

    This method specifies that a division ring has no zero divisors, i.e. is a domain.

    See also

    The Deduction rules section in the documentation of axioms

    EXAMPLES:

    Sage

        sage: DivisionRings().extra_super_categories()
        (Category of domains,)
        sage: "NoZeroDivisors" in DivisionRings().axioms()
        True

    Python

        >>> from sage.all import *
        >>> DivisionRings().extra_super_categories()
        (Category of domains,)
        >>> "NoZeroDivisors" in DivisionRings().axioms()
        True