# Division rings¶

class sage.categories.division_rings.DivisionRings(base_category)

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.

The Deduction rules section in the documentation of axioms

EXAMPLES:

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