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()
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
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