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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