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# Division rings

class `sage.categories.division_rings.``DivisionRings`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_singleton`

  The category of division rings

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

  EXAMPLES:

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

  TESTS:

      sage: TestSuite(DivisionRings()).run()

  `Commutative`

    alias of `Fields`

  class `ElementMethods`
  `DivisionRings.``Finite_extra_super_categories`()

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

    EXAMPLES:

    Any field is a division ring:

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

    TESTS:

        sage: DivisionRings().Finite() is Fields().Finite()
        True

    This works also for subcategories:

        sage: class Foo(Category):
        ....:     def super_categories(self): return [DivisionRings()]
        sage: Foo().Finite().is_subcategory(Fields())
        True

  class `DivisionRings.``ParentMethods`
  `DivisionRings.``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: DivisionRings().extra_super_categories() (Category of domains,) sage: “NoZeroDivisors” in DivisionRings().axioms() True

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